(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 isoccurring, for 0 < 0 < 2n, when 0 =, theminimum 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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(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 0se < 2n, when e = minimum value of D„f is -|Vf, occurring when the direction of u is ---Select- , the 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 - x?y decreases fastest at the point (5, -1).

(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 0se < 2n, when e = minimum value of D„f is -|Vf, occurring when the direction of u is ---Select- , the 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 - x?y decreases fastest at the point (5, -1).

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 24EQ

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