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Let \( f \) be a non-negative function defined on the interval \( [0,1] \). If \( \int_{0}^{x} \sqrt{1-\left(f^{\prime}(…

Q1

Let \( f \) be a non-negative function defined on the interval \( [0,1] \). If \( \int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} d t=\int_{0}^{x} f(t) d t, 0 \leq x \leq 1 \) and \( f(0)=0 \) then

[JEE Main 2021, 31 Aug (Shift 1)]

a

\( f\left(\frac{1}{2}\right)<\frac{1}{2} \) and \( f\left(\frac{1}{3}\right)>\frac{1}{3} \)

b

\( f\left(\frac{1}{2}\right)>\frac{1}{2} \) and \( f\left(\frac{1}{3}\right)>\frac{1}{3} \)

c

\( f\left(\frac{1}{2}\right)<\frac{1}{2} \) and \( f\left(\frac{1}{3}\right)<\frac{1}{3} \)

d

\( f\left(\frac{1}{2}\right)>\frac{1}{2} \) and \( f\left(\frac{1}{3}\right)<\frac{1}{3} \)

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