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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 Duf = |VA cos e 8 v. Since the minimum value of cos e Ov is -1 v occurring, for 0 se< 2x, when 0= 7t the minimum value of Duf is -|Vf), ocurring when the direction of u is the opposite of y 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 (3, -4). Need Help? Watch It Read 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 0 be the angle between Vf(x) and unit vector u. Then Duf = |VA cos e 8 v. Since the minimum value of cos e Ov is -1 v occurring, for 0 se< 2x, when 0= 7t the minimum value of Duf is -|Vf), ocurring when the direction of u is the opposite of y 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 (3, -4). Need Help? Watch It Read 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 0 be the angle between Vf(x) and unit vector u. Then Duf = |Vf cos e Since the minimum value of cos e Ov is -1 V occurring, for 0 se < 2x, when e = 7T v , the minimum value of Duf is -|V, occurring when the direction of u is the opposite of e 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 (3, -4). Need Help? Watch It Read 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 0 be the angle between Vf(x) and unit vector u. Then Duf = |Vf cos e Since the minimum value of cos e Ov is -1 V occurring, for 0 se < 2x, when e = 7T v , the minimum value of Duf is -|V, occurring when the direction of u is the opposite of e 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 (3, -4). Need Help? Watch It Read It
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