Let f be a differentiable function satisfying f(x)=2/√3 ∫₀^(√3) f((λ² x)/3) d λ, x>0 and f(1)=√3. If y=f(x) passes through the point (α, 6), then α is equal to _____________.

Solution

Maths sir · 4 steps

Put t = (λ² x)/3 so x moves into the limit, then differentiate both sides.

1. Change the variable. Let t = (λ² x)/3, so λ = √(3t/x) and dλ = √3/2√x√t dt. The limits λ = 0 and λ = √3 become t = 0 and t = x.

   f(x) = 2/√3 ∫₀ˣ f(t) √3/2√x√t dt ⇒ √x f(x) = ∫₀ˣ f(t)/√t dt

2. Apply Leibniz rule. Differentiate both sides with respect to x: f(x)/2√x + √x f'(x) = f(x)/√x. This gives √x f'(x) = f(x)/2√x, that is, f'(x)/f(x) = 1/2x.
3. Solve the equation. Integrating gives ln f(x) = 1/2 ln x + C, so f(x) = k√x. From f(1) = √3 we get k = √3, so f(x) = √3x.
4. So. √3α = 6 gives 3α = 36, so α = 12.

The idea to keep. When x sits inside the integrand, substitute so that x moves into a limit. Then Leibniz rule turns the integral equation into a simple differential equation.

Answer from the NTA final answer key. Solution by PrepWiser.