Find and classify all the extreme points for the function f(x,y) =𝑥^2 +𝑦^2 -xy+x.?
- PopeLv 71 month agoFavorite Answer
Level curves are in the form x² - xy + y² + x = k, which is an ellipse if it is anything.
That makes the surface z = f(x, y) an elliptic paraboloid. It must have one extremum.
Intersecting it with vertical plane x = 0 results in parabola z = y², which has no maximum. The single extremum must be a minimum.
Find the partial derivatives, and let each of them equal zero.
∂/∂x f(x, y) = 2x - y + 1
∂/∂y f(x, y) = -x + 2y
Solve this system:
2x - y + 1 = 0
-x + 2y = 0
x = -2/3
y = -1/3
f(-2/3, -1/3) = -1/3
Function f has one minimum, -1/3. It has no other extrema.
- rotchmLv 71 month ago
State here the procedure you saw for such problems. Then we will proceed.
Hopefully no one will spoil you the answer thereby depriving you from your personal enhancement; that would be very inconsiderate of them.