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I need to show that if $f: (a,b) \to \mathbb{ R }\text{ with}\;\; f''( x ) \geq 0$ for all $x \in (a,b)$, then $f\left( \frac{ x + y }{ 2 } \right) \leq \frac{ f( x ) + f( y ) }{ 2 }$.

I know that since $f''( x ) \geq 0$, then $f'(x)$ is monotone increasing. I'm not really sure where to go from here.

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