Let \(f:[1, \infty) \rightarrow[2, \infty)\) be a differentiable function. If \(10 \int_{1}^{\mathrm{x}} f(\mathrm{t}) \mathrm{dt}=5 \mathrm{x} f(\mathrm{x})-\mathrm{x}^{5}-9\) for all \(\mathrm{x} \geq 1\), then the value of \(f(3)\) is:

JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Definite Integrals

Let

f : [1, \infty) \to [2, \infty)

be a differentiable function. If

10 \int_{1}^{x} f (t) dt = 5 x f (x) - x^{5} - 9

for all

x \geq 1

, then the value of

f (3)

is:

Answer: (B)

Differentiating

10 \int_{1}^{x} f (t) d t = 5 x f (x) - x^{5} - 9

gives

10 f (x) = 5 f (x) + 5 x f^{'} (x) - 5 x^{4}

, i.e.

x f^{'} (x) - f (x) = x^{4}

, or

f^{'} - \frac{f}{x} = x^{3}

. The integrating factor

\frac{1}{x}

turns this into

(\frac{f}{x})^{'} = x^{2}

, so

\frac{f}{x} = \frac{x ^{3}}{3} + C

. Setting

x = 1

in the original equation gives

5 f (1) - 10 = 0

, so

f (1) = 2

and

C = \frac{5}{3}

. Then

\frac{f ( 3 )}{3} = 9 + \frac{5}{3} = \frac{32}{3}

, so

f (3) = 32

JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Single correct option