Extrema: First Derivative Test Practice

Sample questions

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1. Sample 1difficulty 1/5

   At a local max, $f'$ changes from

   • A

     Negative to positive

   • B

     Positive to positive

   • C

     Positive to negative

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   • D

     Zero to zero

   Why

   Increasing then decreasing → max.

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2. Sample 2difficulty 1/5

   A critical point of $f$ is a value of $x$ in the domain where

   • A

     $f(x) = 0$

   • B

     $f''(x) = 0$

   • C

     $f$ is increasing

   • D

     $f'(x) = 0$ or $f'(x)$ undefined

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   Why

   Critical points: derivative is zero or undefined (interior of domain).

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3. Sample 3difficulty 2/5

   Given $f'(x) = (x-1)(x-3)^2$, $f$ has a local extremum at

   • A

     $x = 1$ only

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   • B

     $x = 3$ only

   • C

     $x = 1$ and $x = 3$

   • D

     Nowhere

   Why

   Sign change of $f'$ only at $x = 1$ ($x = 3$ is a double root, no sign change). $f'$ goes from $-$ to $+$ at $x = 1$ → local min.

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4. Sample 4difficulty 2/5

   On the interval shown, $f'(x) < 0$ on:

   • A

     $(2, 4)$

   • B

     Everywhere

   • C

     $(0, 2)$

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   • D

     Nowhere

   Why

   $f$ decreases from $x=0$ to $x=2$ (the minimum), so $f'<0$ on $(0,2)$.

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5. Sample 5difficulty 2/5

   For $f(x) = x^3 - 3x^2$, $f$ is <strong>increasing</strong> on

   • A

     $(-\infty, 1)$

   • B

     $(-\infty, 0)$ and $(2, \infty)$

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   • C

     $(0, 2)$

   • D

     Everywhere

   Why

   $f' > 0$ for $x < 0$ or $x > 2$.

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