$\"\"$

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$\"\"$ and $\"\"$ are the two functions whose first and second derivatives exist on the entire interval.

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(a) Statement : $\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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

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$\"\"$

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Product rule of derivatives : $\"\"$ .

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$\"\"$

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Hence $\"\"$ .

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The statement is true.

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$\"\"$

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(b) Statement : $\"\"$ .

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Consider $\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$ .

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

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$\"\"$

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$\"\"$ .

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Hence $\"\"$ .

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The statement is false.

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(a) The statement is true.

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(b) The statement is false.