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(a) Show that a differentiable function / decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of -V(x). Let be the angle between Vf(x) and unit vector u. Then Duf = |VA|---Select---✔ direction of u is-Select-- 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) = xy - x2y3 decreases fastest at the point (1, -1). Since the minimum value of ---Select--- is occurring, for 0 ≤0 < 2m, when = , the minimum value of Duf is-IV, occurring when the

(a) Show that a differentiable function / decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of -V(x). Let be the angle between Vf(x) and unit vector u. Then Duf = |VA|---Select---✔ direction of u is-Select-- 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) = xy - x2y3 decreases fastest at the point (1, -1). Since the minimum value of ---Select--- is occurring, for 0 ≤0 < 2m, when = , the minimum value of Duf is-IV, occurring when the

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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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 8 be the angle between Vf(x) and unit vector u. Then Duf = |Vf||---Select--- . Since the minimum value of ---Select--- is direction of u is ---Select--- ✓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) = x¹y x²y³ decreases fastest at the point (1, -1). occurring, for 0 ≤ 0 < 2π, when 8 = the minimum value of Duf is -|Vf, occurring when the
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 8 be the angle between Vf(x) and unit vector u. Then Duf = |Vf||---Select--- . Since the minimum value of ---Select--- is direction of u is ---Select--- ✓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) = x¹y x²y³ decreases fastest at the point (1, -1). occurring, for 0 ≤ 0 < 2π, when 8 = the minimum value of Duf is -|Vf, occurring when the
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