Question

Find the maximum rate of change of f at the given point and the direction in which it occurs. f(p, q) = 8qe−p + 4pe−q, (0, 0) maximum rate of change $$ Correct: Your answer is correct. direction $$

Answer 1

Explanation:

`Given \ f (p,q) = 8qe^(-p) + 4pe^(-q) \ where \ P_o(0,0) \\ \text{thus the gradient of f is: } \\ \\ \bigtriangledown f(p,q) = \Big((\partial f)/(\partial p), (\partial f)/(\partial q) \Big) \\ \\ (\partial f)/(\partial p) = -8qe^(-p) + 4pe^(-q) \\ \\ (\partial f)/(\partial p) = 8qe^(-p) - 4pe^(-q) \\ \\ Then: \bigtriangledown f(p.q) = (-4qe^(-p)+ 8qe^(-q), 4qe^(-p)- 8qe^(-q)) \\ \\ f(0,0) = (-4*(0)e^(-0)+ 8*(0)e^(-(0)), 4*(0)e^(-(0))- 8*(0)e^(-(0))) \\ \\ = (0+8,4-0) = (8.4)`

`\Big| \Big | \bigtriangledown f(0,0) = √((8)^2+4^2) \\ \\ = √(64+16) \\ \\ = √(80)`

`\mathbf{the \ direction \ of \ maximum \ change \ is }= \mathbf{√(80)} \\ \\ \mathbf{direction }(8,4)`