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(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.

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.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).
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