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Theorems for Differentiability

If the Rolle'...

Question

If the Rolle's theorem holds for the function $f (x) = 2 x^{3} + a x^{2} + b x$ in interval [-1, 1] for the point $c = \frac{1}{2}$ , then find the value of $2 a + b$ ?

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Solution

The correct option is A -1
$f (- 1) = f (1)$
$- 2 + a - b = 2 + a + b$
$\Rightarrow b = - 2$
$f^{'} (c) = 6 c^{2} + 2 a c + b$ .
$f^{'} (c) = 0$ at $c = \frac{1}{2}$
$\Rightarrow a = \frac{1}{2}$
$2 a + b = - 1$

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