Mean Value Theorem and Extreme Value Theorem Practice

Sample questions

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1. Sample 1difficulty 2/5

   Rolle's theorem (special case of MVT) states that if $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there is $c$ with

   • A

     $f'(c) = 0$

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   • B

     $f''(c) = 0$

   • C

     $f(c) = f(a)$

   • D

     $f(c) = 0$

   Why

   Rolle's: $f'(c) = 0$.

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2. Sample 2difficulty 2/5

   On a closed interval $[a, b]$, absolute extrema of a continuous $f$ can occur at

   • A

     Critical points only

   • B

     Critical points or endpoints

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   • C

     Endpoints only

   • D

     Inflection points

   Why

   Closed-interval method: check $f$ at all critical points and at the two endpoints; pick the largest/smallest.

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3. Sample 3difficulty 2/5

   The Extreme Value Theorem guarantees absolute max and min for

   • A

     Any function on any interval

   • B

     Differentiable functions on open intervals

   • C

     Polynomial functions on $\mathbb{R}$

   • D

     Continuous functions on closed intervals

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   Why

   EVT: continuous on closed bounded interval ⇒ attains absolute extrema.

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4. Sample 4difficulty 2/5

   The Mean Value Theorem requires $f$ to be

   • A

     Just $f(a) = f(b)$

   • B

     Continuous on $[a,b]$

   • C

     Both A and B

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   • D

     Differentiable on $(a,b)$

   Why

   MVT requires continuity on $[a,b]$ and differentiability on $(a,b)$.

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5. Sample 5difficulty 2/5

   For $f(x) = x^2$ on $(0, 1)$ (open), the absolute extrema

   • A

     Min at $x = 0$, max at $x = 1$

   • B

     Don't exist

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   • C

     Are at $x = 0$ and $x = 1$

   • D

     Both at the midpoint

   Why

   Open interval — endpoints not included; sup and inf approach $0$ and $1$ but are never attained.

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