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Official 2018 Paper 2 · Q18

Paper 2Logic & proofFunctions & graphsAlgebra

f(x)f(x) is a polynomial function defined for all real xx.

Which of the following is a necessary condition for the inequality

f(a)+f(b)2≥f(a+b2)\frac{f(a)+f(b)}2\geq f\left(\frac{a+b}2\right)

to be true for all real numbers aa and bb with a<ba<b?

Af(x)≥0f(x)\geq0 for all real xx
Bf′(x)≥0f'(x)\geq0 for all real xx
Cf′′(x)≥0f''(x)\geq0 for all real xx
Df(x)≤0f(x)\leq0 for all real xx
Ef′(x)≤0f'(x)\leq0 for all real xx
Ff′′(x)≤0f''(x)\leq0 for all real xx

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