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Consider 2 convex functions g(x) and f(x).
Using property of second derivative test we have:
g(x)'' > 0 and f(x)'' > 0.
We are interested in showing that y(x) = g(x) + f(x) is also convex.
y(x)'' = c*g(x)'' + k*f(x)'', where c and k are positive numbers (where is no convex function those coeficients after diferentiation will give you negative numbers).
Because c, g(x)'', k, f(x)'' are > 0 , we get y(x)'' also >0, hence y(x) is a convex function.
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