I have to prove that the sum of convex functions is again convex.
I know the definition of convex function: $f(tx_1+(1-t)x_2)\leq t f(x_1)+(1-t)f(x_2)$ – this the first convex function, then I have the second one $g(tx_1+(1-t)x_2)\leq t g(x_1)+(1-t)g(x_2)$
What should I do next? Thank you for your help and time.
Best Answer
Hint: $$(f+g)(tx_1 +(1-t)x_2)=f(tx_1 +(1-t)x_2)+g(tx_1 +(1-t)x_2)\leq ...$$