$\"\"$

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The function is $\"\"$ and $\"\"$ .

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Rolle $\"\"$ s theorem :

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Let $\"\"$ be a function that satisfies the following three hypotheses.

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1. $\"\"$ is continuous on $\"\"$ .

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2. $\"\"$ is differentiable on $\"\"$ .

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3. $\"\"$ .

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Then there is a number $\"\"$ in $\"\"$ such that $\"\"$ .

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$\"\"$

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The function is $\"\"$ .

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The function $\"\"$ is continuous on the interval $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Rolle $\"\"$ s theorem is applicable.

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$\"\"$

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The function $\"\"$ .

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Differentiating on each side with respect to $\"\"$ .

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Apply Quotient rule in derivatives $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ is differentiable on the open interval $\"\"$ , which satisfies the Rolle $\"\"$ s Theorem.

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$\"\"$

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$\"\"$

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$\"\"$

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Solve the equation $\"\"$ .

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Compare it to $\"\"$

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$\"\"$

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Roots are $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ is in the interval $\"\"$ .

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$\"\"$

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The function $\"\"$ satisfy the rolle $\"\"$ s theorem.

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$\"\"$ .