2. Define a function f(X) = c. X = C₁x₁ + C₂x₂ + C = (C1, C2, ..., Cn) E R is constant. (x1,x2,...,xn) ER", where (a) Show that for any P and Q in R” and t € R, 0 ≤ t ≤ 1, f(tP + (1 − t)Q) = tf(P) + (1 − t)f(Q). Note that tf (P) + (1 – t)f(Q) is a real number between f(P) and f(Q). (b) What does this suggest about the optimal value of f on a convex set S in R? ...Cnn for X =

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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2. Define a function f(X) = c • X = C₁x₁ + C₂X₂ +
= (C1, C2, ..., Cn) € Rn is constant.
C =
Cnn for X (x1,x2,...,xn) € R", where
=
(a) Show that for any P and Q in R” and t € R, 0 ≤ t ≤ 1, f(tP + (1 − t)Q) = tf(P) + (1 −
t)f(Q). Note that tƒ(P) + (1 − t)ƒ(Q) is a real number between ƒ(P) and ƒ(Q).
(b) What does this suggest about the optimal value of f on a convex set S in R?
Transcribed Image Text:2. Define a function f(X) = c • X = C₁x₁ + C₂X₂ + = (C1, C2, ..., Cn) € Rn is constant. C = Cnn for X (x1,x2,...,xn) € R", where = (a) Show that for any P and Q in R” and t € R, 0 ≤ t ≤ 1, f(tP + (1 − t)Q) = tf(P) + (1 − t)f(Q). Note that tƒ(P) + (1 − t)ƒ(Q) is a real number between ƒ(P) and ƒ(Q). (b) What does this suggest about the optimal value of f on a convex set S in R?
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