JEE Mains · Maths · STD 12 - 5. continuity and differentiation
lf \(f(x)\) is a differentiable function in the interval \((0,\infty )\) such that \(f(1) = 1\) and \(\mathop {\lim }\limits_{t \to x} \frac{{{t^2}f(x) - {x^2}f(t)}}{{t - x}} = 1,\) for each \(x > 0,\) then \(f (\frac {3}{2})\) is equal to
- A \(\frac {23}{18}\)
- B \(\frac {13}{6}\)
- C \(\frac {25}{9}\)
- D \(\frac {31}{18}\)
Answer & Solution
Correct Answer
(D) \(\frac {31}{18}\)
Step-by-step Solution
Detailed explanation
\(\lim_{t \to x} \frac{t^2f(x) - x^2f(t)}{t - x} = \lim_{t \to x} \left[ f(x)\frac{t^2 - x^2}{t - x} - x^2\frac{f(t) - f(x)}{t - x} \right]\) \(= f(x)(2x) - x^2f'(x)\) \(2xf(x) - x^2f'(x) = 1\) \(x^2f'(x) - 2xf(x) = -1\) \(\frac{f'(x)}{x^2} - \frac{2f(x)}{x^3} = -\frac{1}{x^4}\)…
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