Answer to Question #170338 in Real Analysis for Prathibha Rose

Question

Show by an example that in general a continuous function is neither convex nor concave

Answer 1

Take the sine function. Then take any two points a,b from $[0,\pi].$ Then in this range the function is convex. The reason is $f(x)=sin \ x\Rightarrow f^{''}(x)=-sin \ x <0.$ Since in the given range the sine function is positive. Again from $[\pi,2\pi ], f^{''}(x)>0$ since in the given range the sin function is negative.Hence in that range the function is concave. Hence in general a continuous function is neither concave or convex.