Question

given the graph of y = f(x) below, at which of the marked x - values can the following statements be true? (for each question, enter your answer as a comma - separated list, e.g., x1,x3,x5. enter none if no points satisfy the given condition) a. f(x) < 0 at x = b. f(x) < 0 at x = c. f(x) is decreasing at x = d. f(x) is decreasing at x = e. the slope of f(x) is positive at x = f. the slope of f(x) is positive at x =

Explanation:

Step1: Analyze when $f(x)<0$

Check where the graph is below the x - axis.

Step2: Analyze when $f'(x)<0$

$f'(x)<0$ when the function is decreasing (slope is negative).

Step3: Analyze when $f(x)$ is decreasing

Same as when $f'(x)<0$, look for downward - sloping parts of the graph.

Step4: Analyze when $f'(x)$ is decreasing

This is when the slope of the tangent to the function is getting less steep in the negative direction (concave down and decreasing).

Step5: Analyze when the slope of $f(x)$ is positive

This is when $f'(x)>0$, i.e., the function is increasing.

Step6: Analyze when the slope of $f'(x)$ is positive

This is when the function is concave up and increasing.

Answer:

A. x4
B. x3, x4
C. x3, x4
D. x2
E. x1, x2, x5
F. x4, x5