How to prove the following f(x) is convex function

Question

1. Verify directly from the definition that the function of one variable f(x) = e^x is convex function.

2. Prove two variables function :f(x1, x2) = x1^2 + 3x2^2 - 3x1x2 + 2x1 is convex

Answer 1

Directly from the definition, a function $f$ is convex if $$f(tx + (1 - t) y) \le tf(x) + (1 -t) f(y)$$ for $0 \le t \le 1$. Therefore, for (1), you should directly check that the inequality $$e^{tx + (1 - t)y} \le te^x + (1 - t)e^y$$ for $0 \le t \le 1$ is true.

For the second part, perhaps you can recall that for smooth functions, there is a certain condition on the second derivative (which is the Hessian, in multiple variables) that guarantees convexity.