Problems
1. Prove that \( f(x, y, z)=(x+2 y+3 z)^{2} \) is convex. (Hint: Use [17.19].)
2. To what extent do parts (a), (c), and (e) of Theorem 17.6 remain true if one considers strictly concave instead of concave functions?
3. Prove Jensen's inequality [17.18] in Section 17.6 for the case in which \( f \) is differentiable by using the following idea: By [17.20], concavity of \( f \) implies that \( f(x(t))-f(z) \leq f^{\prime}(z)[x(t)-z] \). Multiply both sides of this inequality by \( \lambda(t) \) and integrate w.r.t. \( t \). Then let \( z=\int_{a}^{b} \lambda(t) x(t) d t \).
