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Gresty Academy Paper 2 · Q17

Paper 2Logic & proof

For which of the following functions is

f(a)+f(b)2≤f(a+b2)\frac{f(a)+f(b)}2\leq f\left(\frac{a+b}2\right)

true for all real numbers a,ba,b where a<ba<b?

Af(x)=1−x2f(x)=1-x^2
Bf(x)=sin⁡(x)f(x)=\sin(x)
Cf(x)=exf(x)=e^x
Df(x)=e−xf(x)=e^{-x}
Ef(x)=x2−1f(x)=x^2-1

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