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(b) Use the result of part (a) to find the direction in which the function f(x, y) = x³y - x2y³ decreases fastest at the point (2, -4).

(b) Use the result of part (a) to find the direction in which the function f(x, y) = x³y - x2y³ decreases fastest at the point (2, -4).

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 f be a differentiable function and be the angle between Vf(x) and unit vector u. Then Df = |vf|cos(0) -1 ✔ occurring for 0 ≤ 0 < 2π, when = π direction of Vf (assuming Vf is not zero). Since the minimum value of cos(8) ✓ ✓ is , the minimum value of Df is -1Vfl, occurring when the direction of u is the opposite of✔✔✔ (b) Use the result of part (a) to find the direction in which the function f(x, y) = x³y - x²y3 decreases fastest at the point (2, -4). What is the rate of decrease? 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 f be a differentiable function and be the angle between Vf(x) and unit vector u. Then Df = |vf|cos(0) -1 ✔ occurring for 0 ≤ 0 < 2π, when = π direction of Vf (assuming Vf is not zero). Since the minimum value of cos(8) ✓ ✓ is , the minimum value of Df is -1Vfl, occurring when the direction of u is the opposite of✔✔✔ (b) Use the result of part (a) to find the direction in which the function f(x, y) = x³y - x²y3 decreases fastest at the point (2, -4). What is the rate of decrease? the
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