Question

find all points where the function has any relative extrema or saddle points and identity the type of relative extrema
f(x,y) = 7xy
a. saddle point at (0,0)
b. relative maximum at (0,0)
c. relative minimum at (-1, -1), saddle point at (0,0)
d. no relative extrema or saddle points

Explanation:

Step1: Find first - order partial derivatives

The first - order partial derivatives of \(f(x,y)=7xy\) are:
\(f_x=\frac{\partial f}{\partial x}=7y\), \(f_y=\frac{\partial f}{\partial y}=7x\)

Step2: Find critical points

Set \(f_x = 0\) and \(f_y=0\).
From \(f_x = 7y = 0\), we get \(y = 0\). From \(f_y=7x = 0\), we get \(x = 0\). So the critical point is \((0,0)\)

Step3: Find second - order partial derivatives

\(f_{xx}=\frac{\partial^2 f}{\partial x^2}=0\), \(f_{yy}=\frac{\partial^2 f}{\partial y^2}=0\), \(f_{xy}=\frac{\partial^2 f}{\partial x\partial y}=7\)

Step4: Use the second - derivative test

The discriminant \(D=f_{xx}f_{yy}-(f_{xy})^2\)
Substitute \(f_{xx} = 0\), \(f_{yy}=0\), \(f_{xy}=7\) into the formula: \(D=(0\times0)-7^2=- 49<0\)

Answer:

A. Saddle point at \((0,0)\)