sketch a qualitatively accurate graphs of the first derivative f'(x) and the second derivative f''(x) from the graphs of f(x).

Question

Answer 1

graph 1:

f'(x) is the slope of f(x):

starting from left: more (-) to less (-) until zero at x=0, then less (+) to more (+)

so it is a straight line which passes through the (0,0) and with a positive slope. .

f"(x) is the slope of f(x) which is positive and flat.

graph 2:

f'(x):

1st part is a straight line, from more (+) to less (+) but not zero.

2nd part is a straight line, (-), flat, one value

3rd part is straight line, from less (+) to more (+).

f"(x)

1st part is a straight line, flat, (-), one value

2nd part is zero, flat straight line

3rd part is a straight line, flat, (+), one value

Noticed that the absolute value (magnitude) of the (-) and (+) value has the same value.

HIH!

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Answer 2

For the first problem, I've sketched in the slopes on light blue in the sketch below. They start out negative, get less and less negative until the slope is zero at the minimum point. Then they become more and more positive. The slopes are the first derivative and I summarized them with the red line. The slope of the first derivative is the second derivative, which is a constant value shown in green.

Similarly, for the second graph except there are places the derivative does not exist.

Answer 3