Question

Find the direction in which the function f=x^2-y^2+2xy decreases most rapidly at
the point(1, 1).

Answer 1

Answer on Question #46382, Engineering, Other

Problem.

Find the direction in which the function $f = x^2 - y^2 + 2xy$ decreases most rapidly at the point (1, 1).

Solution:

The function increases the fastest in the direction of gradient and decreases the fastest in the opposite direction. Hence the direction equals $-\nabla f(1,1) = -4\vec{i}$, as $-\nabla f = (-2x - 2y)\vec{i} + (2y - 2x)\vec{j}$. Therefore the vector of the direction has coordinates $(-4,0)$.

Answer: $(-4,0)$.

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