Exercise II.5. (a) Which are the following functions (defined from R to R) are convex: (i) f(x) = e (iii) f(x) = (x² - 1)² (iv) fs = ² (ii) fe(x) = 2x²-3x+5

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 58E
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Exercise II.5. (a) Which are the following functions (defined from R to R) are
convex:
(i) f(x) = e
(iii) f(x) = (x² - 1)² (iv) fs = ²
(b) Show each function in Exercise II.1(a) is convex.
(i)
fi
I1
X₂
(ii) fe(x) = 2x²-3x+5
I1
12
(ii) f2 (11)
:= x² + x² - 2
(iii) f3 := 9(x₁ − 1)² + (2+1)²-2 (iv) f₁ = √√√(x₁ + 1)² + (x₂ − 2)²
-
| := 4x² + ² − 1
x1
In
Transcribed Image Text:Exercise II.5. (a) Which are the following functions (defined from R to R) are convex: (i) f(x) = e (iii) f(x) = (x² - 1)² (iv) fs = ² (b) Show each function in Exercise II.1(a) is convex. (i) fi I1 X₂ (ii) fe(x) = 2x²-3x+5 I1 12 (ii) f2 (11) := x² + x² - 2 (iii) f3 := 9(x₁ − 1)² + (2+1)²-2 (iv) f₁ = √√√(x₁ + 1)² + (x₂ − 2)² - | := 4x² + ² − 1 x1 In
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