find all points where the given function has any local extrema. identify any saddle points.\n$f(x,y)=xy + y…
find all points where the given function has any local extrema. identify any saddle points.\n$f(x,y)=xy + y - 12x$\nfind the local maxima. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. there are local maxima located at \n(simplify your answers. type ordered pairs. use a comma to separate answers as needed.\nb. there are no local maxima.
Answer
Explanation:
Step1: Find first - order partial derivatives
The function is (f(x,y)=xy + y-12x). The first - order partial derivatives: (f_x=\frac{\partial f}{\partial x}=y - 12) (f_y=\frac{\partial f}{\partial y}=x + 1)
Step2: Find critical points
Set (f_x = 0) and (f_y=0). From (f_x=y - 12=0), we get (y = 12). From (f_y=x + 1=0), we get (x=-1). So the critical point is ((-1,12)).
Step3: Find second - order partial derivatives
(f_{xx}=\frac{\partial^2 f}{\partial x^2}=0), (f_{xy}=\frac{\partial^2 f}{\partial x\partial y}=1), (f_{yy}=\frac{\partial^2 f}{\partial y^2}=0)
Step4: Use the second - derivative test
The discriminant (D=f_{xx}f_{yy}-(f_{xy})^2) Substitute (f_{xx} = 0), (f_{yy}=0), (f_{xy}=1) into the formula: (D=(0\times0)-1^2=-1<0)
Answer:
B. There are no local maxima.