Let f be a non-negative function defined on the interval [0, 1]. If _ 0 ^ x 1- (f^ (t) )^…
Mathematics · JEE Main · NTA Exams — Integral Calculus
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
- \(f\left(\frac{1}{2}\right)<\frac{1}{2} \text { and } f\left(\frac{1}{3}\right)>\frac{1}{3}\)
- \(f\left(\frac{1}{2}\right)>\frac{1}{2} \text { and } f\left(\frac{1}{3}\right)>\frac{1}{3}\)
- \(f\left(\frac{1}{2}\right)<\frac{1}{2} \text { and } f\left(\frac{1}{3}\right)<\frac{1}{3}\)
- \(f\left(\frac{1}{2}\right)>\frac{1}{2} \text { and } f\left(\frac{1}{3}\right)<\frac{1}{3}\)
Answer
(A) f ( 1 2 ) 1 3
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