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Omni-MATH / For each continuous function f:[0,1]→R, let I(f)=∫₀¹x²f(x) dx and J(x)=∫₀¹x(f(x))² dx. Find the maximum value of I(f)−J(f) over all such functions f.

Problem

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problem

For each continuous function f:[0,1]→Rf: [0,1] \to \mathbb{R}, let I(f)=∫01x2f(x)dxI(f) = \int_0^1 x^2 f(x)\,dx and J(x)=∫01x(f(x))2dxJ(x) = \int_0^1 x \left(f(x)\right)^2\,dx. Find the maximum value of I(f)−J(f)I(f) - J(f) over all such functions ff.
Plain-text mathematical notation (without MathML)
For each continuous function f:[0,1]→R, let I(f)=∫₀¹x²f(x) dx and J(x)=∫₀¹x(f(x))² dx. Find the maximum value of I(f)−J(f) over all such functions f.
Original LaTeX notation
For each continuous function $f: [0,1] \to \mathbb{R}$, let $I(f) = \int_0^1 x^2 f(x)\,dx$ and $J(x) = \int_0^1 x \left(f(x)\right)^2\,dx$. Find the maximum value of $I(f) - J(f)$ over all such functions $f$.

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Source and history

task_id
e3830c02-3ae0-59b7-9446-e98bdaa07ba8
task_key
test--e3830c02-3ae0-59b7-9446-e98bdaa07ba8
task_revision_id
1
config / split
/ test

Official source

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