Activity 16.
(a)
If \(f''\) is positive on an interval, then \(f'\) is \(\underline{\hspace{20mm}}\) on that interval and \(f\) is \(\underline{\hspace{20mm}}\) on that interval.
Answer.
increasing; concave up
(b)
If \(f''\) is negative on an interval, then \(f'\) is \(\underline{\hspace{20mm}}\) on that interval and \(f\) is \(\underline{\hspace{20mm}}\) on that interval.
Answer.
decreasing; concave down
(c)
At exactly two of the labeled points in Figure 154, the derivative \(f'\) is 0; the second derivative \(f''\) is not zero at any of the labeled points. Give the signs of \(f\text{,}\) \(f'\text{,}\) and \(f''\) at each marked point.
A plot of \(f\) with four points labeled.
| Point | \(f\) | \(f'\) | \(f''\) |
|---|---|---|---|
| \(A\) | |||
| \(B\) | |||
| \(C\) | |||
| \(D\) |
Answer.
| Point | \(f\) | \(f'\) | \(f''\) |
|---|---|---|---|
| \(A\) | - | 0 | + |
| \(B\) | + | 0 | - |
| \(C\) | + | - | - |
| \(D\) | - | + | + |
(d)
Sketch graphs of functions satisfying each of the descriptions given below.
-
(I).
The slope is positive and increasing at first, but then is positive and decreasing.
-
(II).
The first derivative of the function whose graph is above.
-
(III).
The second derivative of the first function sketched.
Answer.
Answers vary but (I) should be increasing and concave up at first and then concave down. (II) should be above the horizontal axis and increasing and then decreasing all while being concave down. (III) should be decreasing and passing through through the horizontal axis when the maximum is reached in (II).
