A function is convex if \(\displaystyle f(tx+(1-t)y)\leq t\cdot f(x)+(1-t)f(y)\)
Another way of saying it is that the line that connects \(\displaystyle (x,f(x)), \;\ (y,f(y))\) is above the point on the graph \(\displaystyle \left(tx+(1-t)y, \;\ f(tx+(1-t)y\right)\)
It would appear if you just let t=1/2, then you have it. Then, it's just algebra.
That seem like a good enough idea?.
Make a graph and you can see it.
