$\"\"$

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The equation is $\"\"$ .

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Consider $\"\"$ .

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Consider the interval $\"\"$ .

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Find $\"\"$ :

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$\"\"$

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$\"\"$ .

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Find $\"\"$ :

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$\"\"$

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$\"\"$ .

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Here $\"\"$ and $\"\"$ .

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Hence by intermediate value theorem, thre exist a number $\"\"$ in $\"\"$ , such that $\"\"$ .

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$\"\"$ has a solution on $\"\"$ .

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$\"\"$

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(b)

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Consider $\"\"$ .

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

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$\"\"$

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$\"\"$

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Newtons approximation formula : $\"\"$ .

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$\"\"$ .

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Consider $\"\"$ .

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Substitutte $\"\"$ in $\"\"$ .

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$\"\"$

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Substitutte $\"\"$ in $\"\"$ .

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$\"\"$ .

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Substitutte $\"\"$ in above expression.

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$\"\"$

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Similarly,

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$\"\"$

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Root of the equation is $\"\"$ .

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$\"\"$

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(a)

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$\"\"$ has a solution on $\"\"$ .

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(b)

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Root of the equation is $\"\"$ .