$\"\"$

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The function is $\"\"$ , $\"\"$ .

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(a)

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Graph :

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Sketch the function $\"\"$ over $\"\"$ .

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$\"\"$

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$\"\"$

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(b)

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The slope of the secant line through $\"\"$ and $\"\"$ is

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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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To find the secant line equation, use the point - slope form of line equation.

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The point - slope form of line equation is $\"\"$ .

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Consider one point is $\"\"$ .

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$\"\"$

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Graph :

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Sketch the function $\"\"$ and secant line $\"\"$ .

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$\"\"$

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$\"\"$

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(c)

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The function satisfy the condition for mean value theorem , therefore there exist at least one number $\"\"$ in the $\"\"$ , such that $\"\"$ .

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$\"\"$

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Apply derivative with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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The values of $\"\"$ are $\"\"$ and $\"\"$ .

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The function two tangent line in the given interval.

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First tangent line :

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The point of tangency is $\"\"$ .

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$\"\"$

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The point of tangency is $\"\"$ .

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Slope of tangent $\"\"$ . [Since secant line is parallel to tangent line]

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The point - slope form of line equation is $\"\"$ .

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$\"\"$

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Second tangent line :

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The point of tangency is $\"\"$ .

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$\"\"$

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The point of tangency is $\"\"$ .

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Slope of tangent $\"\"$ . [Since secant line is parallel to tangent line]

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The point - slope form of line equation is $\"\"$ .

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$\"\"$

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Graph :

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Sketch the function $\"\"$ , secant line $\"\"$ and tangent lines $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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(a) Graph of the function $\"\"$ is

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$\"\"$

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(b) The secant line equation $\"\"$ .

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Graph of the secant line $\"\"$ is

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$\"\"$

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(c) The tangent line equations are $\"\"$ and $\"\"$ .

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Graph of the tangent line are $\"\"$ and $\"\"$ is

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$\"\"$ .