Daniel J Smith Custom TMUA Set A Paper 2 · Question 16

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Daniel J Smith Custom TMUA Set A Paper 2 · Q16

Paper 2Logic & proofCalculus

Consider the statement:

if $0\leq f(x)\leq1$ for all real $x$ with $0\leq x\leq1$ , then

(*)\quad\int_0^1 f(x)\,dx\leq\int_0^1(f(x))^2\,dx

Which of the following functions is a counterexample to $(*)$ ?

f(x)=2x

f(x)=x^2+\frac12

f(x)=\frac12

f(x)=1+x

f(x)=0

f(x)=5x(1-x)

f(x)=\sqrt{x}+\frac12

f(x)=1

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