Curve Sketching Practice

Sample questions

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1. Sample 1difficulty 2/5

   For $f(x) = x^3$, the function is

   • A

     Always increasing, with inflection at $x = 0$

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   • B

     Concave up for $x < 0$, concave down for $x > 0$

   • C

     Always decreasing

   • D

     Both A and not C

   Why

   $f'(x) = 3x^2 \ge 0$ — increasing (strictly except at $x = 0$). $f''(x) = 6x$ changes sign at $0$ — inflection.

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2. Sample 2difficulty 3/5

   Given $f' > 0$ on $(-\infty, 2)$, $f' < 0$ on $(2, \infty)$, and $f$ continuous, $f$ has

   • A

     A local maximum at $x = 2$

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   • B

     A local minimum at $x = 2$

   • C

     An inflection point at $x = 2$

   • D

     No critical point

   Why

   $f'$ goes from $+$ to $-$ at $x = 2$ → local max.

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3. Sample 3difficulty 3/5

   $f' < 0$ and $f'' > 0$ on $(a, b)$ describes a graph that is

   • A

     Going up and curving up

   • B

     Going down and curving up (like the right half of a U)

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   • C

     Going down and curving down

   • D

     Going up and curving down

   Why

   Decreasing + concave up — slope is negative but slope is increasing (becoming less negative).

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4. Sample 4difficulty 3/5

   A function with $f(0) = 0$, $f$ increasing on $(0, \infty)$, and concave down everywhere on $(0, \infty)$ — could be

   • A

     $f(x) = -x$

   • B

     $f(x) = \sqrt x$

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   • C

     $f(x) = x$

   • D

     $f(x) = x^2$

   Why

   $\sqrt x$: increases, concave down (since $f'' = -\tfrac{1}{4}x^{-3/2} < 0$).

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