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  3. 6 assume that g is a convex function on rquot...

Question: 6 assume that g is a convex function on rquot...

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(6) Assume that g is a convex function on R, that f is a linear function of a single variable, and in addition that f is a nondecreasing function (which means that f(r) 2 f (s) whenever r 2 s). (a) Show that F fog is convex by directly verifying the convexity inequality F(θε + (1-9)y) < 0F(x) + (1-0)F(y). Explain where each hypothesis (convexity of g, convexity of f, and the fact that f is nondecreasing) is used in your reasoning. (The notation F f o g means that F(x)-f(g()).) (b) Now assume that f and g are both C2. Express the matrix of second derivatives ▽2F(x) in terms of f and g. Prove directly (without using part (a)) that V2F(x) is positive semidefinite at every . Hint: see problem (3) above
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