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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 be the angle between Vf(x) and unit vector u. Then Duf = |vf|| cos 0 Since the minimum value of cos e is -1 occurring, for 0 ≤ 0 < 2π, when = π the minimum value of Duf is -|Vfl, occurring when the direction of u is the opposite of Vf 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) = xy-x2y3 decreases fastest at the point (5, -1). (-10, -25) × Need Help? Read It Watch It

(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 be the angle between Vf(x) and unit vector u. Then Duf = |vf|| cos 0 Since the minimum value of cos e is -1 occurring, for 0 ≤ 0 < 2π, when = π the minimum value of Duf is -|Vfl, occurring when the direction of u is the opposite of Vf 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) = xy-x2y3 decreases fastest at the point (5, -1). (-10, -25) × Need Help? Read It Watch It

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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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 be the angle between Vf(x) and unit vector u. Then Duf = |vf|| cos 0 Since the minimum value of cos e is -1 occurring, for 0 ≤ 0 < 2π, when = π the minimum value of Duf is -|Vfl, occurring when the direction of u is the opposite of Vf 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) = xy-x2y3 decreases fastest at the point (5, -1). (-10, -25) × Need Help? Read It Watch It
Transcribed Image Text:(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 be the angle between Vf(x) and unit vector u. Then Duf = |vf|| cos 0 Since the minimum value of cos e is -1 occurring, for 0 ≤ 0 < 2π, when = π the minimum value of Duf is -|Vfl, occurring when the direction of u is the opposite of Vf 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) = xy-x2y3 decreases fastest at the point (5, -1). (-10, -25) × Need Help? Read It Watch It
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