Question

The graph of f '(x) is continuous and decreasing with an x-intercept at x = -3. Which of the following statements must be true?

Answer 1

When f'(x) > 0, the function is increasing.
Conversely, f'(x) <0, function is decreasing.

Thus the function is increasing in the interval (-∞,-3], and decreasing on [-3,∞). This makes a maximum at x=-3.

Since we do not know anything about f"(x), we do not know about the concavity of the function.

Answer 2

all of them?

f is concave down with a max at x = -3

Answer 3

Since we are given f'(x) is a decreasing function, its derivative must be negative. Hence f ''(x) < 0, and f is concave down for all x.