Let f be a non-negative function defined on the interval [0, 1]. If _0^x 1 - f'(t) ^2 ,dt = _0^x f(t) ,dt, 0 x 1 and f(0) = 0 then

JEE Advanced Mathematics question

Let be a non-negative function defined on the interval . If , and then

(A) and
(B) and
(C) and Correct
(D) and

Answer: (C)

Solution

Differentiate both sides w.r.t. (Leibniz rule): for all . (The book writes this step as ""; the clean justification is differentiation of the identity.)

So , i.e. with ,

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Question ID VK-110720