Question

find all points where the given function has any local extrema. identify any saddle points.
$f(x,y)=xy + y - 12x$
find the local maxima. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. there are local maxima located at
(simplify your answers. type ordered pairs. use a comma to separate answers as needed.
b. there are no local maxima.

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.