# Use the graph below to fill in each entry of the grade with positive, negative or zero? These are the answers?

I got for each category, but I’m told I am wrong. Please help!

### 1 Answer

- PopeLv 72 months ago
Nice image. I did not really expect you to follow up on that earlier attempt.

I like both of the first two columns, so I suppose there is no need to explain them. As for the second derivative, nothing looks right there.

At point A, the concavity of the curve is upward, like a smile, so the second derivative is positive. See the association? Smile/Positive

Skipping to point C, the concavity opens downward, like a frown, so the second derivative is negative. Frown/Negative

Back to point B, the concavity appears to be changing here from upward to downward, from positive to negative. That would be a point of inflection, where the second derivative is zero.

A more rigorous explanation is that the second derivative is the rate of change of the first derivative. Function f is increasing at all three points, which makes the first derivative positive. The reason it hooks upward at A is that the graph is rising at an increasing rate. That is, the first derivative is increasing, so the second derivative is positive. And at B, it is still rising, but at a decreasing rate. The first derivative is decreasing, so the second derivative is negative, and the curve hooks downward.

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