Second Derivative Test Practice

Sample questions

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1. Sample 1difficulty 2/5

   Inflection points occur where

   • A

     $f = 0$

   • B

     $f' = 0$

   • C

     $f''(x) = 0$ always

   • D

     $f''$ changes sign

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   Why

   Inflection: concavity changes — $f''$ changes sign (and $f$ continuous there).

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2. Sample 2difficulty 2/5

   If $f'(c) = 0$ and $f''(c) = 0$, the second-derivative test is

   • A

     Inconclusive — try first-derivative test or higher derivatives

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   • B

     Always gives a max

   • C

     Always gives an inflection point

   • D

     Always gives a min

   Why

   $f''(c) = 0$ at a critical point: cannot conclude max/min from 2nd test alone.

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3. Sample 3difficulty 2/5

   If $f'(c) = 0$ and $f''(c) > 0$, then $c$ is a

   • A

     Inflection point

   • B

     Local minimum

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   • C

     Cannot tell

   • D

     Local maximum

   Why

   $f''(c) > 0$ at a critical point → concave up → local min.

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4. Sample 4difficulty 3/5

   For $f(x) = x^3 - 3x^2$, the inflection point is at

   • A

     $x = 2$

   • B

     No inflection

   • C

     $x = 0$

   • D

     $x = 1$

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   Why

   $f'' = 6x - 6 = 0 \Rightarrow x = 1$. $f''$ changes sign there.

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5. Sample 5difficulty 3/5

   At a critical point where $f''(c) < 0$, $f$ has a:

   • A

     Cannot tell

   • B

     Local minimum

   • C

     Local maximum

     check_circle
   • D

     Inflection point

   Why

   $f'(c) = 0$ and $f''(c) < 0$ implies a local max.

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