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Which of the following functions satisfy the conditions of Rolle's Theorem on the interval $[-1,1]?$ 
$$
\begin{aligned}
&f(x)=1-x^{2 / 3}\\
&g(x)=x^{3}-2 x^{2}-x+2\\
&h(x)=\cos \left(\frac{\pi}{4}(x+1)\right)
\end{aligned}
$$
Rolle's Theorem applies to:

  1. both $f$ and $g$
  2. both $g$ and $h$
  3. $g$ only
  4. $h$ only
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2 Answers

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$f(x)=1-x^{2 / 3}\Rightarrow$  not differentiable at $0.$

$g(x)=x^{3}-2 x^{2}-x+2 \Rightarrow g(-1) = g(1)$

$h(x)=\cos \left(\frac{\pi}{4}(x+1)\right)\Rightarrow  h(-1) \neq h(1)$
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2 votes
2 votes
Rolle’s Theorem states that,

The function f(x) must be continuous in [-1,1], differentiable in (-1,1). and f(-1) = f(1)

F(x) isn’t differentiable at x=0, as f’x = -2/3 x^-1/3 , so f(x) doesn’t satisfy rolle’s theorem.

G(x) satisfy all conditions

H(x) doesn’t satisfy f(-1) = f(1)
Answer:

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