Answer to Question #138341 in Calculus for Promise Omiponle

Question

What are the local maxima, minima, and saddle points of the following function? What are the values of the function at those points? f(x, y) = (1 +xy)(x+y)

Answer 1

Given, function is

$$f(x,y)=(1+xy)(x+y)=x+y+x^2y+y^2x$$

Now, critical points are determined by $\nabla f(x,y)=0$ , thus

$$\nabla f(x,y)=(1+2xy+y^2,1+2xy+x^2)=0\\ \implies 1+2xy+y^2=0,1+2xy+x^2=0\\ \implies y^2=x^2\implies y=\pm x$$

Clearly, $y\ne x$ as $1+3y^2\ne 0,\forall y\in \mathbb{R}$

Thus, we conclude that $y=-x\implies 1-x^2=0\implies x=\pm1$

Therefore critical points are (1,-1) and (-1,1)

Now, we have to check the Hessian matrix for f at critical points to check whether above critical points are maxima, minima or saddle points.

Now,

$$Hf(x,y)=\begin{bmatrix} 2y&2(x+y)\\ 2(x+y)&2x \end{bmatrix}$$

Now, clearly $Hf(1,-1)=\begin{bmatrix} -2&0\\0&2 \end{bmatrix}\&Hf(-1,1)=\begin{bmatrix} 2&0\\0&-2 \end{bmatrix}$

Clearly, both $Hf(1,-1)\&Hf(-1,1)$ are diagonal hence their eigen values are $\pm 2$

Hence, both $Hf(1,-1)\&Hf(-1,1)$ are semi positive definite ,thus (1,-1) and (-1,1) are saddle points.