Determine the local extrema of the function.
-$f(x, y)=x^{2}+x y+y^{2}-3 x+2$
A) $\mathrm{f}(-2,1)=11$, maximum
B) $f(-2,1)=11$, minimum
C) $f(2,-1)=-1$, minimum
D) $f(2,-1)=-1$, maximum

Question

Quizplus · Accepted Answer

To find the local extrema, we first find the critical points by setting the partial derivatives equal to zero. Solving $\frac{\partial f}{\partial x} = 2x + y - 3 = 0$ and $\frac{\partial f}{\partial y} = x + 2y = 0$, we get the critical point $(2, -1)$. Evaluating the second derivative test, we find that this point is a local minimum with $f(2, -1) = -1$.

Determine the Local Extrema of the Function f(x,y)=x2+xy+y2−3x+2f(x, y)=x^{2}+x y+y^{2}-3 x+2f(x,y)=x2+xy+y2−3x+2

Determine the Local Extrema of the Function $f(x, y)=x^{2}+x y+y^{2}-3 x+2$