Determining Concavity Practice

Sample questions

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1. Sample 1difficulty 2/5

   This concave-down (hill-shaped) graph has

   • A

     Cannot tell

   • B

     $f'' = 0$

   • C

     $f'' > 0$

   • D

     $f'' < 0$

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   Why

   Concave down ⇔ $f'' < 0$.

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2. Sample 2difficulty 2/5

   This U-shaped graph is concave up, so

   • A

     $f''(x) = 0$

   • B

     $f''(x) < 0$

   • C

     $f'(x) > 0$

   • D

     $f''(x) > 0$

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   Why

   Concave up ⇔ $f'' > 0$.

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3. Sample 3difficulty 2/5

   For $f(x) = x^4$, $f''(x) = 12 x^2 \ge 0$. So $f$ is

   • A

     Concave up everywhere

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   • B

     Linear

   • C

     Has an inflection at $x = 0$

   • D

     Concave down everywhere

   Why

   $f'' \ge 0$, with equality only at $x = 0$ (no sign change), so concave up everywhere.

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4. Sample 4difficulty 2/5

   For $f$ with $f''(x) > 0$ on an interval, the graph is

   • A

     Linear

   • B

     Decreasing

   • C

     Concave down

   • D

     Concave up

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   Why

   $f'' > 0$ ⇔ $f'$ increasing ⇔ concave up.

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5. Sample 5difficulty 2/5

   For $f(x) = x^3 - 6x^2 + 9x$, on what interval is $f$ concave up?

   • A

     $(-\infty, 1)$

   • B

     $(1, 3)$

   • C

     $(-\infty, 2)$

   • D

     $(2, \infty)$

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   Why

   $f''(x) = 6x - 12 > 0 \Leftrightarrow x > 2$.

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