Question

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

x³-y³- 2xy+2

Short answer

The maximum point of the given function is

Step-by-step solution

Given data

A function is given as .f(x,y)=x³-y³-2xy+2.

Concept of maximum and minimum points of a function

To evaluate the 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.

Differentiate the equation of  f(x , y)

Consider the given function as follows:

f(x , y) = x³ - y³-2xy + 2 ....................(1)

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

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

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

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

Solve for the value of x and  y

Equate equation (2) with zero as follows:

...............(4)

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

.............(5)

Solve equation (4) and (5) to obtain the value of X .

Now solve as shown below to obtain the value of Y.

The obtained points are (0,0) and

Calculation for the minimum points of the function  f( x , y)

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

f_xx( x, y ) = 6x .................(6)

f_{yy ( x, y ) = -6y .......................(7)}

f_xy ( x , y ) = -2y ..................(8)

Now, use the value obtained in equation (6) ,(7) and (8) to check the maximum and minimum point.

Therefore, the point (-2/3,2/3) is maximum point of the given function.