Question

In the following question, find the coordinates of the single stationary point (a,b) of the function f(x,y)=−5x² +2xy+4y² −2x−4y

Answer 1

To find the stationary point of the given function, partial derivatives are set to zero and solved simultaneously, resulting in the stationary point being at coordinates (-1/5, 11/42).

Step-by-step explanation:

The question concerns finding the stationary point of a two-variable quadratic function. The function given is f(x,y) = −5x² + 2xy + 4y² − 2x − 4y. To find the stationary point, we need to use partial derivatives. The partial derivative with respect to x is fx(x,y) = -10x + 2y - 2, and with respect to y is fy(x,y) = 2x + 8y - 4.

To find the stationary point, we set these partial derivatives equal to zero:

1. -10x + 2y - 2 = 0
2. 2x + 8y - 4 = 0

Solving these equations simultaneously gives us the coordinates of the stationary point. Multiplying the second equation by 5 and adding it to the first equation gives:

-10x + 2y - 2 + (10x + 40y - 20) = 0

Simplifying yields 42y - 22 = 0, or y = ⅖. Substituting this into the second original equation gives x = -⅕. Therefore, the stationary point is (a, b) = (-⅕, ⅖).