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13. Suppose a and c are real numbers, c>0, andf is defined on [-1, 1] by (xª sin (|x|-°) (if x + 0), S(x) = (if x = 0). Prove the following statements: (a) f is continuous if and only if a > 0. (b) f'(0) exists if and only if a >1. (c) f' is bounded if and only if a 21+c. (d) f' is continuous if and only if a >1+ c. (e) f"(0) exists if and only if a >2 + c. () f" is bounded if and only if a 22+ 2c. (g) f" is continuous if and only if a > 2+ 2c. 14. Let ƒ be a differentiable real function defined in (a, b). Prove that f is convex if and only if f' is monotonically increasing. Assume next that f"(x) exists for every x € (a, b), and prove that f is convex if and only if f"(x) 20 for all x e (a, b). 15. Suppose a e R', fis a twice-differentiable real function on (a, ), and Mo, are the least upper bounds of f(x)|, \f"(x)|, |f"(x)|, respectively, on (a, 0). M1, M2 Prove that

13. Suppose a and c are real numbers, c>0, andf is defined on [-1, 1] by (xª sin (|x|-°) (if x + 0), S(x) = (if x = 0). Prove the following statements: (a) f is continuous if and only if a > 0. (b) f'(0) exists if and only if a >1. (c) f' is bounded if and only if a 21+c. (d) f' is continuous if and only if a >1+ c. (e) f"(0) exists if and only if a >2 + c. () f" is bounded if and only if a 22+ 2c. (g) f" is continuous if and only if a > 2+ 2c. 14. Let ƒ be a differentiable real function defined in (a, b). Prove that f is convex if and only if f' is monotonically increasing. Assume next that f"(x) exists for every x € (a, b), and prove that f is convex if and only if f"(x) 20 for all x e (a, b). 15. Suppose a e R', fis a twice-differentiable real function on (a, ), and Mo, are the least upper bounds of f(x)|, \f"(x)|, |f"(x)|, respectively, on (a, 0). M1, M2 Prove that

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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Question 13, 14, and 15 please. On question 13, please change f(x) = x^a sin(|x|^-c) to f(x)=|x|^a sin(|x|^-c).

13. Suppose a and c are real numbers, c > 0, and f is defined on [-1, 1] by (x* sin (|x|-) f(x) = to (if x + 0), 'x = 0). Prove the following statements: (a) fis continuous if and only if a> 0. (b) f'(0) exists if and only if a>1. (c) f' is bounded if and only if a21+ c. (d) f' is continuous if and only if a >1+ c. (e) f"(0) exists if and only if a > 2 +c. () f" is bounded if and only if a22+ 2c. (g) f" is continuous if and only if a > 2+ 2c. 14. Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f' is monotonically increasing. Assume next that f"(x) exists for every x € (a, b), and prove that f is convex if and only if f"(x) 20 for all x e (a, b). 15. Suppose a e R', fis a twice-differentiable real function on (a, 0), and Mo, M1, M2 are the least upper bounds of f(x)|, \f"(x)|, \Ff"(x)|, respectively, on (a, o). Prove that Mi<4M, M2.
Transcribed Image Text:13. Suppose a and c are real numbers, c > 0, and f is defined on [-1, 1] by (x* sin (|x|-) f(x) = to (if x + 0), 'x = 0). Prove the following statements: (a) fis continuous if and only if a> 0. (b) f'(0) exists if and only if a>1. (c) f' is bounded if and only if a21+ c. (d) f' is continuous if and only if a >1+ c. (e) f"(0) exists if and only if a > 2 +c. () f" is bounded if and only if a22+ 2c. (g) f" is continuous if and only if a > 2+ 2c. 14. Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f' is monotonically increasing. Assume next that f"(x) exists for every x € (a, b), and prove that f is convex if and only if f"(x) 20 for all x e (a, b). 15. Suppose a e R', fis a twice-differentiable real function on (a, 0), and Mo, M1, M2 are the least upper bounds of f(x)|, \f"(x)|, \Ff"(x)|, respectively, on (a, o). Prove that Mi<4M, M2.
Hint: If h >0, Taylor's theorem shows that 1 f'(x) [f(x + 2h) – f(x)] – hf"($) | 2h for some g e (x, x+ 2h). Hence Мо If"(x)| <hM2 + h To show that M{ = 4M, M2 can actually happen, take a = -1, define (2x² - 1 (-1 <x< 0), f(x)={x? – 1 (0 <x< ∞), x² + 1 and show that Mo = 1, Mı =4, M2=4. Does Mi <4M, M2 hold for vector-valued functions too?
Transcribed Image Text:Hint: If h >0, Taylor's theorem shows that 1 f'(x) [f(x + 2h) – f(x)] – hf"($) | 2h for some g e (x, x+ 2h). Hence Мо If"(x)| <hM2 + h To show that M{ = 4M, M2 can actually happen, take a = -1, define (2x² - 1 (-1 <x< 0), f(x)={x? – 1 (0 <x< ∞), x² + 1 and show that Mo = 1, Mı =4, M2=4. Does Mi <4M, M2 hold for vector-valued functions too?
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