Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(a) Suppose f() has a gradient at x ER". For any v ER", show that f'(x; rv) = rf'(x; v) for any r > 0. This property means the map v→ f'(x; v) is positively homogeneous.

(a) Suppose f() has a gradient at x ER". For any v ER", show that f'(x; rv) = rf'(x; v) for any r > 0. This property means the map v→ f'(x; v) is positively homogeneous.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter14: Discrete Dynamical Systems
Section14.3: Determining Stability
Problem 13E: Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii...
See similar textbooks
icon
Related questions
Question
Exercise II.3. Suppose f: R → R. (a) Suppose f(-) has a gradient at x € R". For any v € R", show that f'(x; rv) = rf'(x; v) for any r > 0. This property means the map v → f'(x; v) is positively homogeneous. (b) Show a differentiable function f() is constant (i.e. there exists c ER with f(x) = cVxER") if and only if Vf(x) = 0 VxR". (Hint: to prove the (←) direction, use the Mean Value Theorem).
Transcribed Image Text:Exercise II.3. Suppose f: R → R. (a) Suppose f(-) has a gradient at x € R". For any v € R", show that f'(x; rv) = rf'(x; v) for any r > 0. This property means the map v → f'(x; v) is positively homogeneous. (b) Show a differentiable function f() is constant (i.e. there exists c ER with f(x) = cVxER") if and only if Vf(x) = 0 VxR". (Hint: to prove the (←) direction, use the Mean Value Theorem).
Expert Solution
steps

Step by step

Solved in 4 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer