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Questions  >>  CBSE XII  >>  Math  >>  Relations and Functions
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Q)

Verify Rolle's theorem for $f(x) = x^2 + 2$ in the interval [-2,2]

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A)
(i)$f(x)= x^2+2$ is a polynomial function,so it is continuous in [-2, 2]
(ii) $\; f ' (x) = 2x,$ this exist in (-2, 2)
$\; \therefore f(x)$ is differentiable in (-2, 2)
(iii) $\; f(-2) = (-2)^2 + 2 = 6$
$\; \; \;f(2) = (2^2 + 2) = 6$
$\implies f(-2) = f(2)$
Hence all the conditions are verified
$f'(x) = 2x \implies f'(c) = 2C$
$\therefore f'(c) = 0 \implies 2C = 0$
$\implies C = 0 \in (-2, 2)$
Hence Rolle's theorem is verified
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