Question

question consider the graph of f(x) below. select all of the following points at which f(x) has a local minimum. select all that apply: x=-1, x=2, x=7/3, x=4, none of the above.

Explanation:

To determine where \( f(x) \) has a local minimum, we use the First Derivative Test: a function \( f(x) \) has a local minimum at \( x = c \) if \( f'(x) \) changes from negative to positive at \( x = c \) (i.e., \( f'(x) < 0 \) for \( x < c \) and \( f'(x) > 0 \) for \( x > c \)).

Step 1: Analyze \( x = -1 \)

• Check the sign of \( f'(x) \) around \( x = -1 \). From the graph, \( f'(x) \) does not change from negative to positive at \( x = -1 \) (likely changes from positive to negative or other, so not a local min).

Step 2: Analyze \( x = 2 \)

• Check the sign of \( f'(x) \) around \( x = 2 \). From the graph, \( f'(x) \) does not change from negative to positive at \( x = 2 \) (maybe positive to negative or flat, not a local min).

Step 3: Analyze \( x = \frac{7}{3} \)

• Check the sign of \( f'(x) \) around \( x = \frac{7}{3} \). From the graph, \( f'(x) \) does not change from negative to positive at \( x = \frac{7}{3} \) (not a local min).

Step 4: Analyze \( x = 4 \)

• Check the sign of \( f'(x) \) around \( x = 4 \). From the graph, \( f'(x) \) changes from negative to positive at \( x = 4 \) (so \( f(x) \) has a local minimum here).

Step 5: Analyze "None of the above"

• Since \( x = 4 \) is a valid local minimum, "None of the above" is incorrect.

Answer:

\( \boldsymbol{x = 4} \) (and check the box for \( x = 4 \))