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Now assume that f and g are both C2. Express the matrix of second derivatives V²F(x) in terms of f and g. Prove directly (without using part (a)) that V²F(x) is positive semidefinite at

Now assume that f and g are both C2. Express the matrix of second derivatives V²F(x) in terms of f and g. Prove directly (without using part (a)) that V²F(x) is positive semidefinite at

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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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) > f(s) whenever r > s). (a) Show that F := fog is convex by directly verifying the convexity inequality F(0x + (1 – 0)y) <OF(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 og mcans that F(x) = f(9(x)).) (b) Now assume that f and g are both C2. Express the matrix of second derivatives V²F(x) in terms of f and g. Prove directly (without using part (a)) that V²F(x) is positive semidefinite at every r. Hint: see problem (3) above. Discussion: Expressing V²F in terms of f and g is basically an exercise in using the chain rule for functions of several variables. If you find it at all difficult, then review the chain rule until you have completely mastered it! When showing that V²F is positive semidefinite, plcase explain again, as you did in part (a), where cach hypothesis is used in your reasoning.
Transcribed Image Text:(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) > f(s) whenever r > s). (a) Show that F := fog is convex by directly verifying the convexity inequality F(0x + (1 – 0)y) <OF(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 og mcans that F(x) = f(9(x)).) (b) Now assume that f and g are both C2. Express the matrix of second derivatives V²F(x) in terms of f and g. Prove directly (without using part (a)) that V²F(x) is positive semidefinite at every r. Hint: see problem (3) above. Discussion: Expressing V²F in terms of f and g is basically an exercise in using the chain rule for functions of several variables. If you find it at all difficult, then review the chain rule until you have completely mastered it! When showing that V²F is positive semidefinite, plcase explain again, as you did in part (a), where cach hypothesis is used in your reasoning.
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