Some simple problems

1. Show that $f(x)=x^2$ is convex for $x \in \mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \in D_f$ and any $\lambda \in [0,1]$ $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
2. Using the second order condition show that the following functions are convex on the specified domain.
• $f(x) = e^x$ is convex for $x \in \mathbb{R}$.
• $g(x) = -\ln(x)$ is convex for $x \in (0,\infty)$.
3. Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \in \mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex.
4. A norm is any function that satisfy the following properties
• $f(\alpha x) = |\alpha| f(x)$ for all $\alpha \in \mathbb{R}$.
• $f(x+y) \leq f(x) + f(y)$
• $f(x) \leq 0$ for all $x \in \mathbb{R}^n$ with equality if and only if $x = 0$

Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).

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