WEBVTT

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- [Voiceover] The figure above
shows the graph of f prime,

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the derivative of a
twice-differentiable function f,

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on the interval, that's a closed interval,

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from negative three to four.

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The graph of f prime
has horizontal tangents

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at x equals negative one, x equals one,

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and x equals three.

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So you have a horizontal
tangent right over,

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a horizontal tangent right over there.

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And let me draw that a little bit neater,

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right over there,

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a horizontal tangent right over there,

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and a horizontal tangent right over there.

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Alright.

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The areas of the regions bounded

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by the x-axis and the graph of f prime

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on the intervals negative two to one,

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closed intervals from negative two to one,

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so this region right over here,

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and the region from one to four,

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so this region right over there,

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they tell us have,

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have the areas are 9 and 12, respectively.

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So that area's 9 and that area is 12.

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So Part A,

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Find all x coordinates at
which f has a relative maximum.

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Give a reason for your answer.

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All x-coordinates at which f

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has a relative maximum.

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So you might say,

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"Oh wait, wait this looks
like a relative maximum

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over here, but this isn't f.

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This is the graph of f prime."

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So let's think about when we used to

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we don't have a graph of f in front of us.

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So let's think about
what it needs to be true

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for f to have a relative
maximum at a point.

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We are probably familiar with

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what relative maximum will look like.

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It'd look like a little lump, like that.

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It could also actually look like that

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but since this is differentiable function

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over the interval, we're
probably not dealing

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with a relative maximum
that looks like that.

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And so what do we know about
a relative maximum point?

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So let's say that's our relative maximum.

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As we approach our relative maximum

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from values, as we have x
values that are approaching

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the x value of our relative maximum point,

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as we approach it from
values below that x value,

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we see that we have a positive slope.

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Our function needs to be increasing.

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So over here.

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Over here we see f is increasing,

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going into the relative maximum point,

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f is increasing,

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which means that the derivative of f

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the derivative of f must
be greater than zero.

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And then after we past that maximum point,

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after we past that maximum point,

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we see that our function
needs to be decreasing.

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Do this in another color.

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We see that our funciton is decreasing

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right over here, so f decreasing,

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decreasing, which means

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that f prime of x needs
to be less than zero.

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So our relative maximum point

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should happen at an x value.

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It should happen at an x value

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where our first derivative transitions

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from being greater than zero

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to being less than zero.

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So what x value,

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so let me say this,

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so we have

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f has relative, let me
just write it shorthand,

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relative maximum at x values

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where

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f prime transitions,

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transitions,

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transitions from

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from positive, positive,

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to negative, to, let me write
this a little bit neater,

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to negative.

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to negative.

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And where do we see f prime transitioning

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from positive to negative?

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Well over here we only
see that happening once.

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We see right here f prime is
positive, positive, positive,

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and then it goes negative,
negative, negative.

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So we see,

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we see f prime is positive over here,

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And then, right when we
hit x equals negative two,

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f prime becomes negative.

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F prime becomes negative.

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So we know that the function itself,

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not f prime,

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f must be increasing here

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because f prime is positive,

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and then our function at f

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is decreasing here

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because f prime is negative.

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And so this happens at x equals two.

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So let me write that down.

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This happens at x equals two.

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This happens,

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happens at x equals two.

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And we're done.