Show that each of the following functions is convex or strictly convex on the specified convex set: (a) f(x₁, x₂) = 5x² + 2x₁x₂ + x² − x₁ + 2x₂ + 3 on D = R². (b) f(x₁, x₂) = x/2 + 3x2/2 + √√3x₁x₂ on D = R². (c) f(x₁, x₂) = (x₁ + 2x₂ + 1)8 − ln(x₁x₂)² on D = {(x₁, x₂) € R²: x₁ > x₂ > 1}. (d) f(x₁, x₂) = 4e³×1-x₂ + Sex ² + x ² on D = R².
Show that each of the following functions is convex or strictly convex on the specified convex set: (a) f(x₁, x₂) = 5x² + 2x₁x₂ + x² − x₁ + 2x₂ + 3 on D = R². (b) f(x₁, x₂) = x/2 + 3x2/2 + √√3x₁x₂ on D = R². (c) f(x₁, x₂) = (x₁ + 2x₂ + 1)8 − ln(x₁x₂)² on D = {(x₁, x₂) € R²: x₁ > x₂ > 1}. (d) f(x₁, x₂) = 4e³×1-x₂ + Sex ² + x ² on D = R².
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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Only a, c, d please
Transcribed Image Text:Show that each of the following functions is convex or strictly convex on the
specified convex set:
(a) f(x₁, x₂) = 5x² + 2x₁ x₂ + x² − x₁ + 2x₂ + 3 on D = R².
(b) f(x₁, x₂) = x/2 + 3x2/2 + √√3x₁x₂ on D
(c) ƒ(x₁, x₂) = (x₁ + 2x₂ + 1)³ − ln(x₁x₂)² on D = {(x₁, x₂) € R²: x₁ > x₂ > 1}.
(d) f(x₁, x₂) = 4e³x1-x₂ +
5ex²+x²
on D = R².
(e) f(x₁, x₂) = C₁X₁ + C₂/X₁ + C3x₂ + C4/X₂ on D =
x₂ > 0} where c; is a positive number for i = 1, 2, 3, 4.
= R².
:{(x₁, x₂) = R²: x₁ > 0,
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