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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) since $f^{\prime}(x)>0$ on $(1,5), f$ is increasing on this interval. since $f^{\prime}(x)<0$ on (0,1) and $(5,6), f$ is decreasing on these

intervals.

(b) since $f^{\prime}(x)=0$ at $x=1$ and $f^{\prime}$ changes from negative to positive there, $f$ changes from decreasing to increasing and has

a local minimum at $x=1 .$ since $f^{\prime}(x)=0$ at $x=5$ and $f^{\prime}$ changes from positive to negative there, $f$ changes from

increasing to decreasing and has a local maximum at $x=5$

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So I've drawn a rough sketch of the graph that we were given which is the derivative of our function F of X. And for part a what we want to figure out where the intervals in which our function is increasing, in the intervals in which our function is decreasing. So where are function is going to be increasing is wherever are derivative is positive. So that's going to be from this point here, when X is equal to one to this point over here, where X is equal to five, since all our values for F prime of X are positive past this 50.1 and before this 0.5. So for two, ever figure out where a function F of X is increasing from its derivative and you're giving the graph of the derivative, Just figure out where we have positive Y values for our derivatives. So are derivative, is increasing increasing on the interval from one 25 And we don't include one or five since we're derivative is equal to zero at those points. And we're not increasing at those points. And now to figure out where we're decreasing, it's going to be wherever our graph is negative. So In this case our graph has negative y values from 0 to 1 and from 5 to 6. So we're decreasing from 0 to 1 and from five 2 6. And again we don't include five. We can actually include six since I'm we have a negative value at that point. So we won't include one since we're zero at that point, we won't include five, but we will include six since we are negative at that point. And now, for part B, what we want to figure out is where our local minimum or maximum values for our function F. Of X. R. Where they're located. So if we look at our graph again, for our derivative of F of X, the maximum and minimum function or minimum values are going to be Located where our function f prime of X is equal to zero. So the two places where we could have a maximum minimum are at this point. And at this point. And to figure out if these are actual maximum or minimum values, we're gonna look at points before and after these points. So before this point we have negative f. Prime of X values. So we're going from negative and then after this point we have positive. So we're going from negative to positive. So we're decreasing and then we're going to be increasing. So if we're looking at a graph that was decreasing or decreasing, decreasing, decreasing, then we then we have a constant or zero slope and then we start increasing. We can see that that would have to be a minimum value, a local minimum value. So wherever our function goes from increasing, they're sorry, from negative values to positive values. That means our function F a bex goes from decreasing the increasing, which means the local minimum. So the local minimum occurs at this point X is equal to one, so local men At X is equal to one. And then if we look at this point at X is equal to five, we can see we're going from increasing or decreasing and so our slope is going from positive to negative. And that means we have a local maximum. So you have a local max At X is equal to five and these don't have to be local. They could also be the actual absolute maximum or absolute minimums. But for there to be min or max is you have to have a change in sign when you're derivative of your function F of X is at zero. So you couldn't have a zero that just touched. Like if we had a different a different F. Prime of X function and it looked like this, we're just touched the X axis and went from positive to positive. That wouldn't be a local backs or local minimum.