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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 (ii) fe(x) = 2x²-3x+5

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

Advanced Engineering Mathematics
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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**Exercise II.5.** **(a)** Which of the following functions (defined from ℝ to ℝ) are convex: (i) \( f_5(x) = e^x \) (ii) \( f_6(x) = 2x^2 - 3x + 5 \) (iii) \( f_7(x) = (x^2 - 1)^2 \) (iv) \( f_8 = e^{x^2} \) **(b)** Show each function in Exercise II.1(a) is convex. --- **Functions for Part II.1(a):** (i) \( f_1 \left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) := x_1^2 + x_2^2 - 2 \) (ii) \( f_2 \left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) := 4x_1^2 + \frac{x_2^2}{9} - 1 \) (iii) \( f_3 \left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) := 9(x_1 - 1)^2 + \frac{(x_2 + 1)^2}{4} - 2 \) (iv) \( f_4 \left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) := \sqrt{(x_1 + 1)^2 + (x_2 - 2)^2} \) **Explanation/Notes:** - Each function \( f_1, f_2, f_3, \) and \( f_4 \) is a mapping from \(\mathbb{R}^2\) to \(\mathbb{R}\). - Convexity refers to a property of functions where the line segment between any two points on the graph of the function lies above or on the graph. - For the functions defined from \(\mathbb{R}\) to \(\mathbb{R}\), you are tasked with determining which are convex and proving the convexity of any given functions in part (b).
Transcribed Image Text:**Exercise II.5.** **(a)** Which of the following functions (defined from ℝ to ℝ) are convex: (i) \( f_5(x) = e^x \) (ii) \( f_6(x) = 2x^2 - 3x + 5 \) (iii) \( f_7(x) = (x^2 - 1)^2 \) (iv) \( f_8 = e^{x^2} \) **(b)** Show each function in Exercise II.1(a) is convex. --- **Functions for Part II.1(a):** (i) \( f_1 \left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) := x_1^2 + x_2^2 - 2 \) (ii) \( f_2 \left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) := 4x_1^2 + \frac{x_2^2}{9} - 1 \) (iii) \( f_3 \left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) := 9(x_1 - 1)^2 + \frac{(x_2 + 1)^2}{4} - 2 \) (iv) \( f_4 \left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) := \sqrt{(x_1 + 1)^2 + (x_2 - 2)^2} \) **Explanation/Notes:** - Each function \( f_1, f_2, f_3, \) and \( f_4 \) is a mapping from \(\mathbb{R}^2\) to \(\mathbb{R}\). - Convexity refers to a property of functions where the line segment between any two points on the graph of the function lies above or on the graph. - For the functions defined from \(\mathbb{R}\) to \(\mathbb{R}\), you are tasked with determining which are convex and proving the convexity of any given functions in part (b).
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