Chapter 3: Q. 65 (page 249)

Prove Rolle’s Theorem: If $f$ f is continuous on $[a, b]$ and differentiable on $(a, b)$ , and if $f (a) = f (b) = 0$ , then there is some value $c \in (a, b)$ with $f (c) = 0$ .

Short Answer

Expert verified

We have proved the statement is true for Rolle's Theorem.

Step by step solution

Step 1. Given Information.

The function $f$ is given.

Step 2. Rolle's Theorem.

If $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ with $f (a) = f (b) = 0$ . We know that $f$ attains both a maximum and a minimum value on $[a, b]$ . If one of these extreme values occur at $x = c$ in the interior $(a, b)$ of the interval, then $x = c$ is a local extremum of $f$ . Using the theorem of "Local Extrema are Critical Points" this means that $x = c$ is a critical point of $f$ . Hence, it follows that $f' (c) = 0$ as $f$ is assumed to be differentiable at $x = c$ . Hence, Rolle's Theorem is verified.

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Prove Rolle’s Theorem: If $f$ f is continuous on $[a, b]$ and differentiable on $(a, b)$ , and if $f (a) = f (b) = 0$ , then there is some value $c \in (a, b)$ with $f (c) = 0$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Rolle's Theorem.

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Prove Rolle’s Theorem: If ff is continuous on [a,b]and differentiable on (a,b), and if f(a)=f(b)=0, then there is some value c∈(a,b)with f(c)=0.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Rolle's Theorem.

One App. One Place for Learning.

Most popular questions from this chapter

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Calculus

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Study anywhere. Anytime. Across all devices.

Prove Rolle’s Theorem: If $f$ f is continuous on $[a, b]$ and differentiable on $(a, b)$ , and if $f (a) = f (b) = 0$ , then there is some value $c \in (a, b)$ with $f (c) = 0$ .