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Understanding The Extreme Value Theorem, Critical Numbers, Absolute Minimum & Maximum
Feb 19, 2025
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We will be discussing the Extreme Value Theorem, Critical Numbers, Absolute Minimum & Maximum. This theorem is important in Calculus and Analysis. It basically states that given a function defined on a closed interval, there must exist a maximum and minimum value of the function on that interval. We will be discussing what these terms mean and how to find them.
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0:00
look at the ball here
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so I have three
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separate functions right and I want you
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to recognize some patterns here so if
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you go from right to left so what do you
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see here
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about this function
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absolute value right it looks like a V
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what would you call this point
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it's the highest point right
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maximum maximum it's a maximum right
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it's a maximum
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right we have a maximum here
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and what you call this here
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there's nothing that's a whole right so
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the function is undefined so there's
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nothing there right we don't have
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anything there and this is just another
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point on the graph correct and what
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would you say about this function
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it is only one a closed animal right
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between A and B do you see that now
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second what do you see here
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any observations
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that's not defined point right it's the
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open envelope so it's open right it
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doesn't end here so we're gonna have
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parenthesis here we're going to have a
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bracket
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right
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and what else do you see here there's a
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function of a maximum
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does it it doesn't
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it's a circle that inside so open circle
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it doesn't end here right it's not a
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close animal what's the point under that
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if he was close though there would be a
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maximum it's an open circle
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so there's no maximum there there's a
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hole right
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you see that now the last one what do
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you notice
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what do you have here
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yeah the function is on the closed
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interval right
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it's a bracket
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then what do you have here what would
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you call this here
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minimum right
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minimum
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and then this will be a
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you use the term maximum or in this case
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since it looks like a parabola a Vertex
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right but it's a maximum right
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so now what can you establish
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about a maximum and a minimum on a
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closed angle what would you say
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maximum is the highest points let's play
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well that's obvious right but for that
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to happen
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there has to be two conditions that have
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to be met right the function has to be
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what first continuous
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right continuous because here's it's
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continuous and then what else
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it has to be bounded right it has to be
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a closed animal does that make sense so
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if
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so this is called the mean
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mean value
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right mean value term in a sense that if
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is a continuous function
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on
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the closed
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interval
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a b
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then we conclude that the conclusion is
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foreign
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right does that make sense so if f is a
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function and F is a continuous function
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on a closed interval so this is always
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going to be true f is going to have both
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a maximum n minimum value now in this
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case the close the interval is closed
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but what the function is not what it's
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not continuous right there's a gap here
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there's a hole here so it's not
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continuous so therefore F only has a
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maximum it does not have a minimum in
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case number two the function
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heart is not on a close animal it's on
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an open animal right A and B are not
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included it's continuous but it's not on
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a close animal and then the third one
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you have a and b it is closed and then
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the function is continuous completion
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and it's both
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a maximum and minimum value does that
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make sense
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any question on that
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all right so now we're going to jump
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onto from that to like relative maximum
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and relative extremas now
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if you have a function give you a
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different function here
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did you guys do um maximums and critical
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numbers in pre-calculus
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or not not at all yes no all right
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so let me give you this function here
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right
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so what you have here is not a problem
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right it's a function
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but then what do you notice about this
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function
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the other possible one a possible
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maximum here right
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so maximum possible
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Max
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and this is what
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a possible
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minimum
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right and then another possible Max
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so what can we draw from this we can say
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that this is an open interval because
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there's no there's no end right it does
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not end anywhere it keeps going from
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negative Infinity right to positive
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Infinity does that make sense
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this is not a closed antibody function
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it keeps going all the way to Infinity
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all the way to Infinity here so what are
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some conclusions that we can make here
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we can call this a relative maximum and
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this is also what we call a relative
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minimum meaning on this on on the entire
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open interval you have the possible
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minimum here you have a possible maximum
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and you have another possible maximum so
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those are called relative
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it depends on how you define that
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function right so if I were to cut this
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function from here to here this will be
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a maximum if I have to cut this function
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from here to here this will be a maximum
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and this will be a minimum so those are
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called relative extremas okay relative
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relative
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extrema
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you know people usually say truth is
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relative truth is relative meaning your
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truth is not my truth so a relative
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maximum is basically one possible
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maximum does that make sense and a
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relative minimum is one possible minimum
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okay
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so you have a relative maximum
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and any relative
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minimum
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this is on on open
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interval okay that happens now
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this is the case with parabolas this is
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the case with like uh sometimes those
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absolute value function you may have a
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relative Max relative mean now we're
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going to go from there to set up uh what
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we call uh critical numbers and then
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derivatives again we're back to the
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rivers again
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so did you write this down good all
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right that are you good
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so now we're going to talk about
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critical numbers and relative
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experiments so critical numbers
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so how do you find a critical number and
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what is a critical number so let's say
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we have a function let's find a function
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here that'll help us real quick
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so if I have a function f of x
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equals to 3x 34 minus
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uh
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4X cubed
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over the integral
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negative one
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to two and the question is
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um find the absolute Max and the
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absolute mean of this function now a
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natural Max is the highest number on on
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the function for a function and an
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absolute mean is the lowest value on the
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function on a given closed animal right
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so to find the absolute Max and the
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absolute mean we have to use what we
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call the critical numbers right so how
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do you find the critical numbers you
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first find the derivative of the
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function find the derivative so what is
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the derivative of this function
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all right
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what's the river
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yes is it 12 x to the Third
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minus 12x to the
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second power second power so that's the
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that's his first derivative now to find
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the critical numbers you're going to set
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the first derivative to zero and solve
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for x and those values are called
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critical numbers so we set this equal to
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zero set the derivative
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equals to zero so we have 12 x cubed
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minus 12 x squared is equal to zero and
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then we're going to solve for x right
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I'm gonna put 12x Square as a factor
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I'm going to be left with x minus 1
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is equal to zero right so my critical
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numbers are going to be I'm going to
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have x squared equal to 0 and then x
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squared minus 1 is equal to zero
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so I have X is equal to zero and then X
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is equal to one so those are called the
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critical numbers
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to find the critical numbers you set the
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derivative equal to zero and you solve
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for x yes
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well 12 times 12 are just divided by 12
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so it's gone right if I have 12x squared
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you don't it doesn't really matter
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because I'm going to divide by 12.
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12 right so X is equal to zero
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so the critical numbers miracle
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numbers
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are
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0 and 1. now how do we use the critical
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numbers so here's what we're going to do
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right on this function here they want us
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to find the absolute maximum and the
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absolute minimum of this function now
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that we found the critical numbers we're
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going to use the critical numbers and
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we're also going to use the end point of
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the closed interval to see which one of
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these values yield to the highest number
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and yield to the lowest numbers and
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those are going to be called absolute
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mean and absolute Max respectively so
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the next step now that we have the
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critical numbers so we're going to take
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each value here so we have
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negative 1 right is the end point and
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I'm zero
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and then one and two so we're going to
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evaluate the function at those given
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points
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and the one that gives us the highest
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value is going to be called the absolute
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Max and the one that gives us the lowest
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is going to be called
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the absolute mean so find f of negative
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one you can you guys can use your
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calculator find F of zero
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find F of one and then find F of 2.
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so obviously F of 0 that's going to give
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us zero right just plug in 0 here so
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that's going to be zero so what's f of
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negative one f of negative one will be
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three
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negative one to the fourth minus four
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negative one to the three
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three and this start with three right
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negative four and negative one that'll
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be positive four
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value positive seven
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and then F of one will give you three
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one to the fourth minus four one to the
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third that is three minus four that's
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negative one and then F of two I need
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you you guys to help me with f of two
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every three two three four
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minus four two to the third what's that
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gonna give you
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what do you get for f of two
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you do it
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this is
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three is sixteen right and then minus
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four times two and two and two that's
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eight
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so that would be 3 and 16 is what
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48 right minus 32 so that is uh 16. all
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right so look here now we have 7
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negative one eight on and then sixteen
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and zero so which one do you think is
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the absolute Max what value gives you
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the absolute match here
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yes
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two right so this is absolute Max
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is a two
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and what's the absolute mean
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foreign
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gives us the absolute minimum
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anyone
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F of one right absolute minimum
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so basically
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to find an absolute minimum and the
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absolute minimum minimum the actual
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maximum of function on the on the closed
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interval
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you value the functions at the end point
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right these are the end point and then
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you find the critical numbers and
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evaluate the function at all those
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values and the one that gives you the
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lowest value is the absolute name and
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the one that gives you the highest value
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is the absolute Max any question on that
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any questions
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no good so now we're going to work on
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some problems in the book