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The expression is $\"\"$ .

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(a)

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Intermediate value theorem :

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The function $\"\"$ is continuous on the closed interval $\"\"$ , let $\"\"$ be the number between $\"\"$ and $\"\"$ , where $\"\"$ then exist a number $\"\"$ in $\"\"$ such that $\"\"$ .

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Consider the function is $\"\"$ .

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Consider the function $\"\"$ to be continuous over the interval $\"\"$ .

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Prove that the number $\"\"$ exists between $\"\"$ and $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ in the function.

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in the function.

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$\"\"$

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Thus, $\"\"$ .

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Then according to intermediate value theorem, there exist atleast one root between $\"\"$ and $\"\"$ such that $\"\"$ .

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(b)

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Consider $\"\"$ be the root exist between $\"\"$ and $\"\"$ .

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Then $\"\"$ .

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$\"\"$ .

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Using calculator the value of $\"\"$ is $\"\"$ .

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Therefore the interval is considered as $\"\"$ .

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Interval of the function containing the root is $\"\"$ .

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Interval of the function containing the root is $\"\"$ .