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

(a) On what intervals is $ f $ increasing? Explain.

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

(c) On what intervals is $ f $ concave upward or concave downward? Explain.

(d) What are the $ x $-coordinates of the inflection points of $ f $? Why?

a) $(0,4),(6,8)$

b) Local Max: $x=4,8$ Local Min: $x=6$

c) $f(x)$ is concave upward on $(\leftarrow, 1),(2,3)$ and $(5,7)$

$f(x)$ is concave downward on $(1,2),(3,5)$ and $(7, \rightarrow)$

d) $x=1, x=2, x=3, x=5$ and $x=7$

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when a graph of the derivative or function F. Of X. And for part a we want to use this graph to figure out where our function F of X is increasing, so f of X is increasing. And then I'll put the interval after. But so to actually figure out this interval, we just have to look at our graph and figure out wherever we have positive Y values. So we have positive Y values from this point to this point here at X is equal to four, so from X is equal to 02 X equal to four. We have positive F prime of X values, which means our slope is positive at those points, which means we're increasing so We're increasing from 0 to 4 and then we're also positive or F prime of X is positive from the value X is equal to or X is equal to 62 X is equal to eight. So we have positive values here. So this also means we're increasing from 6 to 8. And again, the reason that um where are f prime of X is positive means we're increasing is because that's where we have a positive slope and whenever we have a positive slope that means we are increasing. Yeah. And now we're going to figure out where we have local minimum and maximum values. So those occur at our zeros. So the zeros of our derivative tell us where are minimum maximum values are, which are the values that I have circled. And for a value to be a maximum value, we have to go from positive to negative or from an increasing slope to a decreasing slope. And that occurs at this point X is equal to four. And then also at this point X is equal to eight. So we have local max, uh X is equal to four and that X is equal to eight. And the reason that um this is true is because when we go from a positive slope to a negative slope, that means we go from increasing to decreasing, which I can illustrate here. So we're increasing, increasing, increasing. And then we're at a point where we have a slope of zero or a horizontal um slope and then we go for from increasing to decreasing. So that's that's an illustration of going from increasing to decreasing, which then shows us that this is indeed a maximum local maximum value. So, so our local maximums occur at X equal to foreign X equal to eight. And then local minimums are at this other value where we have zero, which is at X is equal to six. And that's because we're going from negative to positive or from a decreasing slope or sorry, from a negative slope to a positive slope, which means that we have a local minimum at that point. And now we want to figure out where we are concave up or concave down. So to figure this out, if we were given the function F of X, we would just figure out where our Second derivative of that function is equal to zero. And then we could go from there, however, were just given the graph of our first derivative, and so we're gonna have to figure out where our second derivative is equal to zero given this graph. And those values, the values at which um our second derivative is going to be equal to zero are at our maximum minimum values. So at this point, at this point, at this point, and then at this point in this point and now I've circled all the points, I'm actually going to use a different color. So we use this color. So green circle points are maximum or minimum values which tell us where we're going from. Concave up to concave down or from concave down to concave up. So if we look at our graph before this point, which is a maximum, we have a positive slope and then we go to a negative slope. So we're going from positive to negative. So at any maximum value of our derivative, that's going to be an inflection point that goes from concave up to concave down. So we know that we're concave up from this point to this maximum and then we're concave down from this max to this minimum. And then we're concave up from this meant to this maximum and then we're concave down From this max all the way to this minimum and so on. So we're concave up from 0 to 1 And then from 2 to 3 and then from 5 to 7. So those are three and a intervals. So concave up from 0 to 1, You 2- three. And this last interval was 5-7. And the concave down intervals are going to be the other intervals of our graph. So we're concave down from If we look at this from 1 to 2 And then from 3 to 5 and then lastly from 7 to 8. And now for part D, we're going to figure out what the X coordinates of our inflection points are. So inflection points are going to be wherever we're going from, concave up to concave down or from concave down to concave up. So any change in common cavity is an inflection point and those are going to be at our maximum minimum values. So the inflection points are going to be X equal to one. X is equal to two. X is equal to three. X is equal to five and X is equal to seven. So PSR inflection points at X equals 1235 and seven. And again, the reason that I know this is because our the derivative of this graph or our second derivative or function f X is going to be zero at these maximum minimum values. And it's also going to go from positive to negative or from negative to positive instead of going from positive to positive or negative to negative. So to illustrate this, the graph of our the derivative of our derivative or the second derivative cf double prime of X at these maximum and minimum values. So at this maximum we're gonna go from positive to negative. So we're going to go it's going to look like this and then at this minimum we're going to go from negative to positive, so we're gonna go up above and then negative again and so on. So at each of these points here on our graph of F double prime, these are going to be inflection points since we go from positive to negative values. And there also is a zero at this actual point. So That is why our inflection points are at 1, 2, 3, 5 and seven. Mhm.