7)

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Step 1:

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The function is $\"\"$ on $\"\"$ .

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The Mean Value Thereom:

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If $\"\"$ is continuous on the closed interval $\"\"$ and differentiable on the open interval $\"\"$ , then there exists a number $\"\"$ in $\"\"$ such that $\"\"$ .

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The function $\"\"$ is continuous on $\"\"$ and diifferentiable on $\"\"$ .

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In this case $\"\"$ .

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Step 2:

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Find $\"image\"$ .

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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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Substitute the values of $\"\"$ and $\"\"$ in $\"image\"$ . $\"\"$

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$\"\"$ .

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Step 3:

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The function satisfies the mean value theorem hypotheses, then there exists a number $\"\"$ in $\"\"$ such that $\"\"$ .

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$\"\"$

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute the value of $\"image\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Solution:

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$\"\"$ .

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