(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, inthe direction of -Vf(x).Let be the angle between Vf(x) and unit vector u. Then Duf = |vf|| cos 0✓Since the minimum valueof cos 0is -1occurring, for 0 ≤ 0 < 2л, when =πthe minimumvalue of Duf is -|Vfl, occurring when the direction of u is the opposite of vVf is not zero).the direction of Vf (assuming-(b) Use the result of part (a) to find the direction in which the function f(x, y) = x²y − x²y³ decreases fastest at the point(5, -1).(810, 1053)Need Help?Read ItWatch It

(a) The function f decreases most rapidly at point x in the direction opposite to the gradient…

Answered: (a) Show that a differentiable function…

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 15E

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(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of -Vf(x). Let 0 be the angle between Vf(x) and unit vector u. Then D„f = |Vf| Select--- v . Since the minimum value of -Select--- v is occurring, for 0 < 0 < 2n, when 0 = , the minimum value of D,f is -|Vf|, occurring when the direction of u is -Select--- v the direction of Vf (assuming Vf is not zero). (b) Use the result of part (a) to find the direction in which the function f(x, y) = x*y – x²y³ decreases fastest at the point (5, -1).
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