Intermediate Value Theorem Practice

Sample questions

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1. Sample 1difficulty 2/5

   The Intermediate Value Theorem requires which hypothesis?

   • A

     $f'(a) > 0$

   • B

     $f$ is differentiable on $[a, b]$

   • C

     $f$ is continuous on $[a, b]$

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   • D

     $f(a) = f(b)$

   Why

   IVT requires only continuity on a closed interval — then $f$ takes every value between $f(a)$ and $f(b)$.

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2. Sample 2difficulty 3/5

   For $f(x) = x^3 + x - 1$ with $f(0) = -1$ and $f(1) = 1$, the IVT guarantees a $c \in (0, 1)$ with

   • A

     $f(c) = -1$

   • B

     $f(c) = 1$

   • C

     $f(c) = 0$

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   • D

     $f'(c) = 0$

   Why

   $f$ is continuous and changes sign on $[0,1]$. By IVT there is a $c$ with $f(c) = 0$ between $-1$ and $1$.

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