Question

Let f be a differential function satisfying
f(x) =\(\frac{ 2}{√3} \)\(∫^{√30} f(\frac{λ2x}{3})dλ,x>0 and f(1) = √3.\)
If y = f(x) passes through the point (α, 6), then α is equal to _____

Answer 1

Correct Answer: 12

∵ f(x) = \(\frac{2}{√3} \)\(∫^{√3}_0 f(\frac{λ^2x}{3})dλ, x>0....(i)\)
On differentiating both sides w.r.t., x, we get

f'(x) =\(\frac{2}{√3} \)\(∫^{√3}_0\)\(\frac{λ^2}{3}f'( \frac{λ^2x}{3})dλ\)
f'(x) = \(\frac{1}{√3} \)\(∫^{√3}_0\) λ. \(\frac{λ^2}{3}f'( \frac{λ^2x}{3})dλ\)
\(xf'(x) =\frac{ f(x)}{2}\)
On integrating we get :
In y = \(\frac{1}{2} \)In x + In c
∵ f(1) = √3 then c = √3
∴ (α,6) lies on
∴ y = √3x
∴ 6 = √3α
⇒ α = 12