Let for some function y = f(x), and f(2) = 3. Then f(6) is equal to [2025]

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Solution

Q.
Let for some function y = f(x), $\int_{0}^{x} t f (t) d t = x^{2} f (x), x > 0$ and f(2) = 3. Then f(6) is equal to [2025]

1 3

2 6

3 1

4 2

Ans.
(3)

We have, $\int_{0}^{x} t f (x) d t = x^{2} f (x), x > 0$

Differentiate the given equation, we get

$x f (x) = 2 x f (x) + x^{2} f' (x)$

$\Rightarrow x f (x) = - x^{2} f' (x) \Rightarrow f (x) = - x f' (x)$

$\Rightarrow \int \frac{f' (x)}{f (x)} d x = - \int \frac{1}{x} d x \Rightarrow \log f (x) = - \log x + \log C$

$\Rightarrow \log f (x) = - \log x + \log C \Rightarrow f (x) = \frac{C}{x}$

We have, $f (2) = 3; f (2) = 3 = \frac{C}{2} \Rightarrow C = 6$

So, $f (x) = \frac{6}{x} \Rightarrow f (6) = \frac{6}{6} = 1$

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