If $ f(x) $ is a differentiable function such that $ \int f(x) d x=2[f(x)]^{2}+C $, (where, $ C $ is the constant of integration) and $ f(1)=1 / 4 $, then $ f(\pi) $ equals

Question

If $ f(x) $ is a differentiable function such that $ \int f(x) d x=2[f(x)]^{2}+C $, (where, $ C $ is the constant of integration) and $ f(1)=1 / 4 $, then $ f(\pi) $ equals, $ \frac{3}{4} $, $ \frac{\pi}{4} $, $ \frac{2}{3} $, $ \frac{7}{6} $

MARKS App · Accepted Answer

Differentiating w.r.t. ‘ x ’ we get f x = 4 f x · f ′ x f ′ x = 0 or 1 4 = f x Integrating, we have ∫ 1 4 d x = ∫ f ′ x d x ⇒ x 4 = f x + C As f 1 = 1 4 → C = 0 ⇒ f x = x 4 ⇒ f π = π 4

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