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Integration of Piecewise Continuous Functions

If C1,C2 are ...

Question

If $C_{1}, C_{2}$ are the values of $x, (C_{1} < C_{2})$ for which LMVT holds for the function $f (x) = x^{3}$ on the interval $[- 3, 3],$ then the value of $4 C_{1}^{2} + 7 C_{2}^{2}$ is equal to

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Solution

$f (x) = x^{3}, x \in [- 3, 3]$
$f$ is continuous in $[- 3, 3]$ and differentiable on $(- 3, 3)$
So, from LMVT, $f^{'} (c) = \frac{f (3) - f (- 3)}{3 - (- 3)}$
$\Rightarrow 3 c^{2} = 9$
$\Rightarrow c = \pm \sqrt{3} \in (- 3, 3)$
$∴ C_{1} = - \sqrt{3}, C_{2} = \sqrt{3}$
$\Rightarrow 4 C_{1}^{2} + 7 C_{2}^{2} = 33$

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