Question

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.

Answer 1

Because $f'(x)$ is increasing, you know that $f''(x)>0$, which means $f$ 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 $f'(x)=0$, and you know that the graph of $f'(x)$ crosses the x-axis at $x=0$, so this is also true.

That leaves B. Why is it false? Inflection points occur at points where $f''(x)=0$, where the sign of $f''(x)$ changes to either side of $x$. But you know that $f''(x)>0$ is always true, so there is no such $x$ that makes $f''(x)=0$.

Answer 2

Answer

B is False

Step-by-step explanation:

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