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Official 2016 Paper 2 · Q18

Paper 2Logic & proofCalculusFunctions & graphs

Consider this statement about a function f(x)f(x):

(*) If (f(x))2≤1 for all −1≤x≤1 then ∫−11(f(x))2 dx≤∫−11f(x) dx\text{(*) If }(f(x))^2\leq1\text{ for all }-1\leq x\leq1\text{ then } \int_{-1}^1(f(x))^2\,\dd x\leq\int_{-1}^1f(x)\,\dd x

Which one of the following functions provides a counterexample to (*)?

Af(x)=x+12f(x)=x+\frac12
Bf(x)=x−12f(x)=x-\frac12
Cf(x)=x+x3f(x)=x+x^3
Df(x)=x−x3f(x)=x-x^3
Ef(x)=x2+x4f(x)=x^2+x^4
Ff(x)=x2−x4f(x)=x^2-x^4

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