Final answer:
To find the stationary point of the function f(x, y), we calculate its partial derivatives, set them to zero, and solve the resulting system of equations, resulting in the approximate coordinates of the stationary point being (-0.095, 0.524).
Step-by-step explanation:
The student asked to find the coordinates of the single stationary point (a, b) of the function: f(x, y) = -5x² + 2xy + 4y² - 2x - 4y. To find the stationary points, we first need to calculate the partial derivatives of the function with respect to both x and y and set them equal to zero, which will give us a system of equations to solve.
The partial derivative with respect to x is: f_x(x, y) = -10x + 2y - 2, and with respect to y is: f_y(x, y) = 2x + 8y - 4.
Now we set the partial derivatives to zero to find the stationary points:
- -10x + 2y - 2 = 0 (1)
- 2x + 8y - 4 = 0 (2)
By solving this system of equations, we can find the values of x and y that are the coordinates of the stationary point.
Solving equation (1) for y gives us: y = 5x + 1. Substituting this into equation (2), we get:
2x + 8(5x + 1) - 4 = 0
Which simplifies to:
2x + 40x + 8 - 4 = 0
42x + 4 = 0
x = -1/10.5
Substituting x back into the equation for y we get:
y = 5(-1/10.5) + 1
y = -5/10.5 + 10.5/10.5
y = 5.5/10.5
So the stationary point is (-1/10.5, 5.5/10.5), or approximately (-0.095, 0.524).