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Integral of f(lambda x) over lambda equals k f(x): f is a power of x

This type of question, asked 3 times: 27 Jul 2022, 29 Jan 2025, 24 Jan 2026 S1. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    27 Jul 2022

    178
    Let ff be aa differentiable function satisfying f(x)=23∫03f(λ2x3)dλ,x>0f(x) =\frac{2}{\sqrt{3}}\int_{0}^{\sqrt{3}} f\left( \frac{\lambda^{2}x}{3} \right)d\lambda,x > 0 and f(1)=3f(1) =\sqrt{3}. If y=f(x)y = f(x) passes through the point (α,6)(\alpha,6), then α\alpha is equal to

  2. Question 2

    29 Jan 2025

    71
    Let f:(0,∞)→Rf:(0,\infty) \rightarrow R be a twice differentiable function. If for some a≠0,∫01f(λx)dλ=af(x)a \neq 0,\int_{0}^{1} f(\lambda x)d\lambda = af(x), f(1)=1f(1) = 1 and f(16)=18f(16) =\frac{1}{8}, then 16−f′(116)16 -f^{'}\left( \frac{1}{16} \right) is equal to ____\_\_\_\_ ,

  3. Question 3

    24 Jan 2026 S1

    71
    Let a differentiable function ff satisfy the equation ∫036f(tx36)dt=4αf(x)\int_{0}^{36} f\left( \frac{tx}{36} \right)dt = 4\alpha f(x). If y=f(x)y = f(x) is a standard parabola passing through the points (2,1)(2,1) and (−4,β)( - 4,\beta), Then βα\beta^{\alpha} is equal to ____\_\_\_\_.