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.
find the local minima. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. there are local minima located at
(simplify your answers. type ordered pairs. use a comma to separate answers as needed.
b. there are no local minima.

Explanation:

Step1: Find first - order partial derivatives

The first - order partial derivatives of \(f(x,y)=xy + y-12x\) are:
\(f_x=y - 12\) (derivative with respect to \(x\), treating \(y\) as a constant)
\(f_y=x + 1\) (derivative with respect to \(y\), treating \(x\) as a constant)

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

\(f_{xx}=0\) (second - derivative of \(f\) with respect to \(x\) twice)
\(f_{yy}=0\) (second - derivative of \(f\) with respect to \(y\) twice)
\(f_{xy}=1\) (first - derivative with respect to \(x\) then \(y\))

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:

Since \(D=-1<0\) at the critical point \((-1,12)\), there are no local maxima (B) and no local minima (B). The point \((-1,12)\) is a saddle point.