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Calculus › Extrema and curve sketching › Bounds from derivatives

Minimum ratio of derivative energy to squared mean

University · ●●●●○ 100 pts

Let ff be a continuously differentiable real-valued function on [0,1][0,1] with ∫1/32/3f(x) dx=0\int_{1/3}^{2/3} f(x)\,dx = 0. Find the minimum value of ∫01(f′(x))2 dx(∫01f(x) dx)2\dfrac{\int_0^1 (f'(x))^2\,dx}{\left(\int_0^1 f(x)\,dx\right)^2} over all such ff for which the denominator is nonzero.

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calculus of variationsintegral inequalitiesdefinite integralsintegration by partsoptimization