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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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