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The graph of a function $y=f(x)$ is shown. At which point(s) are the following true?

(a) $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ are both positive.

(b) $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ are both negative.

(c) $\frac{d y}{d x}$ is negative but $\frac{d^{2} y}{d x^{2}}$ is positive.

a) Point $\mathrm{B}$

b) Point $\mathrm{E}$

c) Point $\mathrm{A}$

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Missouri State University

Oregon State University

Harvey Mudd College

So the graph of the function Reichel's aftereffects is shown. So this is effort Bank, by the way, on were asked about Which point are the following statement? True. So we'LL go through each of the statement. So for a we're told that where the slope is positives or do ideas and first driven and it is Khan cave up So d square second derivative is positive. So we want positive slope and and con cave up Well, can't you up occurred right here in this region So an A and B and B has a a positive flow but they have the negative slopes of the answer would be big and then for B So for the next one is telling us that a slowpoke and the second the first and second riveted or both Negative. But I mean, it's con cave down and negative slow. So we have conquered where conclave down at CD and Eve because you could see the upside down new shape right here. And so now we just have determined where the point is. Ah, the slope is negative. So see us positive. Because if you draw a line right here, so positive love this will be a zero slow and he has a slightly negative slope. So it is e and then C is hard to see. But it's telling us that the first group of negative but the second derivative is positive. So we're looking for where it is time to give up less air B and it occurs and the slope is negative. So this world was negative and be so let's just be and that is it.