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MATH / The function f:R→R satisfies

Problem

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problem

The function f:R→Rf : \mathbb{R} \to \mathbb{R} satisfies x2f(x)+f(1−x)=−x4+2xx^2 f(x) + f(1 - x) = -x^4 + 2xfor all real numbers x.x. Then f(x)f(x) can be uniquely determined for all values of x,x, except f(α)f(\alpha) and f(β)f(\beta) for some real numbers α\alpha and β.\beta. Compute α2+β2.\alpha^2 + \beta^2.
Plain-text mathematical notation (without MathML)
The function f:R→R satisfies
x²f(x)+f(1−x)=−x⁴+2xfor all real numbers x.  Then f(x) can be uniquely determined for all values of x, except f(α) and f(β) for some real numbers α and β.  Compute α²+β².
Original LaTeX notation
The function $f : \mathbb{R} \to \mathbb{R}$ satisfies
\[x^2 f(x) + f(1 - x) = -x^4 + 2x\]for all real numbers $x.$  Then $f(x)$ can be uniquely determined for all values of $x,$ except $f(\alpha)$ and $f(\beta)$ for some real numbers $\alpha$ and $\beta.$  Compute $\alpha^2 + \beta^2.$

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
248ee68d-3b56-518b-b2df-be3be5dea89c
task_key
intermediate~5falgebra--test--248ee68d-3b56-518b-b2df-be3be5dea89c
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import