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

∵ 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