The graph of f ′(x) is continuous and increasing with an x-intercept at x = 0.
which of the following statements is false? (4 points)

a. the graph of f is always concave up.
b. the graph of f has an inflection point at x = 0.
c. the graph of f has a relative minimum at x = 0.
d. the graph of the second derivative is always positive.

Because is increasing, you know that , which means must be concave upward over its domain, so A is strue.

By the same fact above, you also know that D must be true.

Relative extrema occur for points where , and you know that the graph of crosses the x-axis at , so this is also true.

That leaves B. Why is it false? Inflection points occur at points where , where the sign of changes to either side of . But you know that is always true, so there is no such that makes .