Question

Find the maximum and minimum values of `f(x,y)=x^3+3xy^2-15x^2-15y^2+72x`

Answer 1

given : 

`f(x,y)=x^3+3xy^2-15x^2-15y^2+72x` 

`f_x=3x^2+3y^2-30x+72  f_(x x)=6x-30`

`f_y=6xy-30y                 f_(y y)=6x-30`

`f _xy=6y`

To find stationary values : 

`f_x=3x^2+3y^2-30x+72=0  &  f_y=6_xy-30y=0`

`y=0  or  x=5 `

for`  x=5  , y=1,-1` 

∴` (x,y)=(5,1),(5,-1)`

Stationary points are : ` (6,0),(4,0),(5,1),(5,-1)`

(i) for point` (6,0),`

`r=f_(x x)= -6, s=f_(xy)=0, f_(yy)=-6`

`rt-s^2=36> 0 and r= 6>0`

function is minimum at (6,0). 

`f_min=108`

(ii) for point (4,0)

`r=f_(x x)=-6, s=f_(x y)=0,t=f_(y y)-6`

`rt-s^2=36>0  and  r= -6<0`

function is maximum at (4,0). 

`f_max=112` 

(iii) for point (5,1) and (5,-1) , 

Thr points are neither maximum nor minimum. 

∴ The maximum and minimum value of function are 112 and 108 .