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Question

The graph of f ′(x) is continuous and increasing with an x-intercept at x = 0. Which of the following statements is false?

Options:

1)	The graph of f is always concave up.
2)	The graph of f has an inflection point at x = 0.
3)	The graph of f has a relative minimum at x = 0.
4)	The graph of the second derivative is always positive.

Answer 1

Answer 2 is false. Let's go down the answer choices.

If the graph of f' is always increasing, then f''(x) must always be positive. This eliminates choices 1 and 4, because if f''(x) is positive then the graph of f must be concave up.

If f'(x) is always increasing, and it has an x-intercept at x=0, then f'(x) must have gone from negative to positive at x=0. By definition, this means that a relative minimum must exist at x=0 on the graph of f, which eliminates choice 3.

This leaves choice 2. An inflection point occurs on the graph of f only if the graph of f''(x) changes signs. However, we know from eliminating choice A that f''(x) is always positive, because f'(x) is always increasing. Therefore, choice 2 is false.