Question

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 \( z=\int_{a}^{b} \lambda(t) x(t) d t \).

          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 \( z=\int_{a}^{b} \lambda(t) x(t) d t \).

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) ≤ f^'(z)[x(t)-z]. Multiply both sides of this inequality by λ(t) and integrate w.r.t. t. Then let z=∫a^bλ(t) x(t) d t.

Added by Edward B.

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