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Let f be a differentiable function satisfying f(x) = $\frac{2}{\sqrt[]{3}} \int_{0}^{\sqrt[]{3}} f (\frac{λ^{2} x}{3}) d λ,$ x > 0 and f(1) = $\sqrt[]{3}$ . If y = f(x) passes through the point (a, 6), then a is equal to………….

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Vishal Baghel | Contributor-Level 10

2 months ago

$f (x) = \frac{2}{\sqrt[]{3}} \int_{0}^{\sqrt[]{3}} f (\frac{λ^{2} x}{3}) d λ$

Substitute $\frac{λ^{2} x}{3} = t$ $\frac{^{}}{}$

$f (x) = \frac{1}{\sqrt[]{x}} \int_{0}^{x} \frac{f (t)}{\sqrt[]{t}} d t$

differentiate using leibneitz rule

$\sqrt[]{x} . f' (x) + f (x) . \frac{1}{2 \sqrt[]{x}} = \frac{f (x)}{\sqrt[]{x}}$

f (1) = $\sqrt[]{3} \Rightarrow f (x) = \sqrt[]{3 x}$

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Let f be a differentiable function satisfying f(x) = $\frac{2}{\sqrt[]{3}} \int_{0}^{\sqrt[]{3}} f (\frac{λ^{2} x}{3}) d λ,$ x > 0 and f(1) = $\sqrt[]{3}$ . If y = f(x) passes through the point (a, 6), then a is equal to………….

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Let f be a differentiable function satisfying f(x) = 2 3 ∫ 0 3 f ( λ 2 x 3 ) d λ , x > 0 and f(1) = 3 . If y = f(x) passes through the point (a, 6), then a is equal to………….

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Let f be a differentiable function satisfying f(x) = $\frac{2}{\sqrt[]{3}} \int_{0}^{\sqrt[]{3}} f (\frac{λ^{2} x}{3}) d λ,$ x > 0 and f(1) = $\sqrt[]{3}$ . If y = f(x) passes through the point (a, 6), then a is equal to………….