QJEEM23Ma4c21: JEE Main 2023 Mathematics - Class 12 JEEM23Ma4c21 Solution: Let f be a continuous function satisfying ∫ 0 t 2 ( f ( x ) + x …

Question

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QJEEM23Ma4c21: JEE Main 2023 Mathematics - Class 12 JEEM23Ma4c21 Solution: Let f be a continuous function satisfying ∫ 0 t 2 ( f ( x ) + x …

Let $f$ be a continuous function satisfying $\int_{0}^{t^{2}} (f (x) + x^{2}) d x = \frac{4}{3} t^{3}, \forall t > 0$ . Then $f (\frac{π^{2}}{4})$ is equal to :

Multiple Choice Options

A −π(1+π316)
B π(1−π316)
C −π2(1+π216)
D π2(1−π216)

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Accepted Answer

Given that

\int_{0}^{t^{2}} (f (x) + x^{2}) d x = \frac{4}{3} t^{3}, \forall t > 0

On differentiating using Leibnitz rule, we get

\begin{aligned} (f (t^{2}) + t^{4}) \times 2 t = \frac{4}{3} \times 3 t^{2} \\ \Rightarrow f (t^{2}) + t^{4} = 2 t \\ \Rightarrow f (t^{2}) = 2 t - t^{4} \end{aligned}

On substituting

\frac{\pi}{2}

for

t

, we get

f (\frac{π^{2}}{4}) = 2 (\frac{π}{2}) - \frac{π^{4}}{16} = π (1 - \frac{π^{3}}{16})

QJEEM23Ma4c21: JEE Main 2023 Mathematics - Class 12 JEEM23Ma4c21 Solution: Let f be a continuous function satisfying ∫ 0 t 2 ( f ( x ) + x …

Multiple Choice Options

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