To find the maximum and the minimum points of the given function.

x² - y²+ 2x - 4y + 10

A function is given as f(x,y) = x²-y²+ 2x - 4y + 10

To evaluate maximum and minimum points of a function f( x , y )the following points should be noted:

are of opposite sign.

Here, f_x, f_y are the first order partial derivatives and f_{xx ,}f_yy, f_xy are the second order partial derivatives.

Consider the given function as f( x , y ) = x² - y² + 2x - 4y + 10 ……. (1)

Differentiate (1) partially with respect to x as shown below:

..........(2)

Differentiate (1) partially with respect to y as shown below:

...........(3)

Equate equation (2) with zero as follows:

Now, equate equation (3) with zero as follows:

The obtained point is (-1,-2).

Now, differentiate (2) and (3) again as shown below:

f_xx ( x , y ) = 2 ............( 4 )

f_yy ( x ,y ) = -2 ............( 5 )

f_{xy ( x , y ) = 0 .................( 6 )}

_{Now, use the value obtained in equation (4),(5) and (6) to check the maximum and minimum point.}

Therefore, the point (-1 , -2) is neither minimum nor maximum point of the given function.