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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|---Select--- . Since the minimum value occurring, for 0 ≤ 0 < 2л, when 0 = of ---Select--- is Duf is -|Vf, occurring when the direction of u is ---Select--- zero). I the minimum value of the direction of Vf (assuming Vf is not (b) Use the result of part (a) to find the direction in which the function f(x, y) = x³y - x²y4 decreases fastest at the point (1, -3).

(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|---Select--- . Since the minimum value occurring, for 0 ≤ 0 < 2л, when 0 = of ---Select--- is Duf is -|Vf, occurring when the direction of u is ---Select--- zero). I the minimum value of the direction of Vf (assuming Vf is not (b) Use the result of part (a) to find the direction in which the function f(x, y) = x³y - x²y4 decreases fastest at the point (1, -3).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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|[ ---Select-- ; . Since the minimum value of ---Select--- ÷ is occurring, for 0 < 0 < 2n, when 0 = the minimum value of Duf is -|Vf|, occurring when the 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) = x³y – ײYA decreases fastest at the point (1, -3).
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|[ ---Select-- ; . Since the minimum value of ---Select--- ÷ is occurring, for 0 < 0 < 2n, when 0 = the minimum value of Duf is -|Vf|, occurring when the 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) = x³y – ײYA decreases fastest at the point (1, -3).
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