Inflection Points of an Even Function

Question

Let $$f$$ be a twice-differentiable function on the interval $$-2 < x < 2$$ such that $$f(x) = f(-x)$$. The table above shows the signs of $$f$$, $$f'$$, and $$f''$$ on intervals within $$0 < x < 2$$. What are the $$x$$-coordinates of the points of inflection of the graph of $$f$$ on $$-2 < x < 2$$?

| | $$0 < x < 1$$ | $$1 < x < 2$$ |
|---|---|---|
| $$f(x)$$ | Positive | Negative |
| $$f'(x)$$ | Negative | Negative |
| $$f''(x)$$ | Negative | Positive |

Accepted Answer

$$x = -1$$ and $$x = 1$$

Answer

There are no points of inflection on $$-2 < x < 2$$.

Answer

$$x = 1$$ only

Answer

$$x = 0$$ only

	$0 < x < 1$	$1 < x < 2$
$f(x)$	Positive	Negative
$f'(x)$	Negative	Negative
$f''(x)$	Negative	Positive

Inflection Points of an Even Function

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