Answer:
The function has a relative minimum value of f(x,y) = -40 at (x,y) = (2,-5) and no relative maximum.
Explanation:
f(x,y) = x² + y² − 4x + 10y − 11
Find the partial derivatives:
∂f/∂x = 2x − 4
∂f/∂y = 2y + 10
Set to 0 and solve:
0 = 2x − 4
0 = 2y + 10
x = 2, y = -5
Evaluate the function at (2,-5):
f(2,-5) = (2)² + (-5)² − 4(2) + 10(-5) − 11
f(2,-5) = 4 + 25 − 8 − 50 − 11
f(2,-5) = -40
Now find the second partial derivative and evaluate at (2,-5):
∂²f/∂x² = 2 > 0
∂²f/∂y² = 2 > 0
The function has a relative minimum value of f(x,y) = -40 at (x,y) = (2,-5) and no relative maximum.