Question

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

Answer 1

Answer on Question #45841 – Math - Vector Calculus

Problem:

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

Solution:

The direction of the fastest decrease is given by the anti-gradient. So let's find the gradient of $f$:

$$f_{x}^{\prime} = (x^{2} - y^{2} + xy)_{x}^{\prime} = 2x + y$$ $$f_{y}^{\prime} = (x^{2} - y^{2} + xy)_{y}^{\prime} = -2y + x$$ $$\operatorname{grad}(f) = (2x + y, x - 2y)^{T}$$

At the point (1,1) this is:

$$\left. \operatorname{grad}(f) \right|_{x = 1, y = 1} = (2 \cdot 1 + 1, 1 - 2 \cdot 1) = (3, -1)$$

Then the fastest decrease is in the reverse direction (anti-gradient) $(-3,1)$.

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