Question

3.
no calculator
the graph of $f$, the derivative of $f$, is shown in the figure above. which of the following describes all relative extrema of $f$ on the open interval $(a,b)$?
(a) one relative maximum and two relative minima
(b) two relative maxima and one relative minimum
(c) three relative maxima and one relative minimum
(d) one relative maximum and three relative minima
(e) three relative maxima and two relative minima

Explanation:

Step1: Recall Extrema from Derivative

A relative extremum of \( f \) occurs where \( f'(x) \) changes sign (by the First Derivative Test). A relative maximum of \( f \) occurs when \( f'(x) \) changes from positive to negative. A relative minimum of \( f \) occurs when \( f'(x) \) changes from negative to positive.

Step2: Analyze \( f'(x) \) Graph

• First sign change: Left of the origin, \( f'(x) \) goes from negative (below x - axis) to positive (above x - axis) → relative minimum of \( f \).
• Second sign change: Near the origin, \( f'(x) \) goes from positive to negative → relative maximum of \( f \).
• Third sign change: To the right of the origin, \( f'(x) \) goes from negative to positive → relative minimum of \( f \). Wait, no, let's re - examine the graph. Wait, the graph of \( y = f'(x) \): starts below x - axis (negative), crosses x - axis (becomes positive), has a peak, then crosses x - axis (becomes negative), has a trough, then crosses x - axis (becomes positive) again.

So the sign changes:

1. From negative to positive (at the first x - intercept) → relative minimum of \( f \).
2. From positive to negative (at the second x - intercept) → relative maximum of \( f \).
3. From negative to positive (at the third x - intercept) → relative minimum of \( f \). Wait, but the options: Option A says one relative maximum and two relative minima. Let's count again.

The graph of \( f'(x) \):

• First interval (leftmost): \( f'(x)<0 \) (negative).
• Then it crosses x - axis (becomes positive): sign change from - to + → relative min of \( f \).
• Then it goes positive, reaches a peak, then crosses x - axis (becomes negative): sign change from + to - → relative max of \( f \).
• Then it goes negative, reaches a trough, then crosses x - axis (becomes positive): sign change from - to + → relative min of \( f \). Wait, but that's one relative max and two relative mins. Which matches option A.

Wait, maybe I miscounted earlier. Let's list the sign changes:

• At the first x - intercept (left of O): \( f'(x) \) goes from - to + → relative min of \( f \).
• At the second x - intercept (between O and the right - most intercept): \( f'(x) \) goes from + to - → relative max of \( f \).
• At the third x - intercept (right of the trough): \( f'(x) \) goes from - to + → relative min of \( f \). Wait, no, that's two relative minima? Wait, no, the graph of \( f'(x) \): let's see the number of sign changes. The graph of \( f'(x) \) crosses the x - axis three times? Wait, no, looking at the graph: starts below x - axis, comes up, crosses x - axis (1st), goes up, then down, crosses x - axis (2nd), goes down, then up, crosses x - axis (3rd). So three x - intercepts.

Sign changes:

1. 1st x - intercept: - to + → relative min of \( f \).
2. 2nd x - intercept: + to - → relative max of \( f \).
3. 3rd x - intercept: - to + → relative min of \( f \). Wait, that's one relative max and two relative mins. So the number of relative maxima is 1, number of relative minima is 2. Which is option A.

Answer:

A. One relative maximum and two relative minima