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The graph of the derivative $ f' $ of a function $ f $ is shown.

(a) On what intervals is $ f $ increasing or decreasing?

(b) At what values of $ x $ does $ f $ have a local maximum or minimum?

a) $f(x)$ is increasing on $(0,1)$ and $(5,7)$

$f(x)$ is decreasing on $(1,5)$ and $(7,8)$

b) $f(x)$ has a local maximum at $x=1$ and $x=7$

$f(x)$ has a local minimum at $x=5$

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Campbell University

Oregon State University

Harvey Mudd College

Idaho State University

So we're given this graph of the derivative or function F. Of X. And we want to figure out where our function is increasing, decreasing. And then we're gonna have to figure out where local minimums and maximum values occur. So for the first part for part a or figure out where we're increasing first. So we're increasing wherever our derivative is positive, so that's going to be from 0-1 and then from 5-7. So we're increasing increasing from 0 to 1 and from five 2, 7, and we don't include zero, we don't include one, we don't include five and we don't include seven because we have a value of zero at those points and that's zero, that's kind of an endpoint of our derivatives. So we're not going to include that point either. And now to figure out where we are decreasing that is wherever our derivative is negative. So if we look at our graph again, that's going to be from 1 to 5 And from 7 to 8. So wherever our function F prime of X. In this case, wherever f prime of X is negative or wherever we have negative Y values is going to be where we're decreasing. So from 1 to 5 and from 7 to 8. And now for part B we want to find where our function has a local minimum and where they have local maximums. So that's going to be where are derivative is equal to zero. So the potential points will be here here and here, so that X is equal to one, X is equal to five and X is equal to seven. And for one of these points to be a local maximum, we have to be going from increasing to decreasing, we have to have a positive slope and then a negative slope. So the only place here that that occurs is going to be actually there are two places where this occurs is at this point. One, since we have a positive slope and then we go to a negative slope And at this .7, since again we have a positive F prime of X or a positive slope for our function F of X. And then it goes to negative f prime of X values. So we have max is at X is equal to one And at X is equal to seven. And then our other point at X is equal to five. That one is going to be a local minimum. Since we're going from negative F prime of X values to positive F prime of X values. We're going from a negative slope for our function F of X to a positive slope. So we're going to be decreasing. Then we're going to have this point at X is equal to five, where we have a slip of zero and then we're going to be increasing. So to illustrate that it would look something like this. If we had a graph for decreasing, decreasing decreasing, we have this point where we're at a zero slope and then we start increasing. So that's going to be a minimum. Either a local minimum maximum or, sorry, a local minimum or an absolute minimum. It's going to be a minimum value at this point X is equal to five, so Minimum at X is equal to five.