Find maximum and minimum values of x^{3} +3xy^{2} -15x^{2}-15y^{2}+72x.

#### Solution 1

Given f(x) = x^{3} +3xy^{2} -15x^{2}-15y^{2}+72x… (1)

STEP 1] for maxima, minima,`(delf)/(delx)=0;(delf)/(dely)=0`

3x^{2}+3y^{2}-30x+72=0 and 6xy – 30y=0

∴ y (6x-30) =0

y=0, x= 5

For x=5; From Equation 3x^{2}+3y^{2}-30x+72=0, we get y2-1=0

Y=±1

Hence (4,0) , (6,0) , (5,1) ,(5,-1) are the stationary points.

STEP 2] Now,`r=(del^2f)/(delx^2)=6x-30;`

`"S"=(del^2f)/(delxdely)=6y;`

`t=(del^2f)/(dely^2)=6x-30`

STEP 3] for (x, y) ≡ (4, 0), r = -6, s = 0, t = -6;

rt –s^{2}=(-6)(-6)-0=36>0 and r< 0.

This shows that the function is maximum at (4, 0)

∴ From Equation (1)

F_{max}=f (4, 0) =4^{3}+0-15(4^{2}) +0+72(4) =64 – 240 + 288

F_{max}=112

STEP 4] For (x,y)≡(6,0)

r=6, s=0, t=6

rt-s^{2}=36 but r=6>0

This shows that function is minimum at (6, 0).

From Equation (1),

F_{min}=f(6,0)=6^{3}+0-15(6)^{2}+0+72(6)=108.

STEP 5] For(x, y) ≡ (5, 1)

r=0, s=6, t=0;

(rt-s^{2})<0

This shows that at (5, 1) and (5,-1) function is neither maxima nor minima.

These points are saddle points.

#### Solution 2

Given f(x) = x^{3} +3xy^{2} -15x^{2}-15y^{2}+72x… (1)

STEP 1] for maxima, minima,`(delf)/(delx)=0;(delf)/(dely)=0`

3x^{2}+3y^{2}-30x+72=0 and 6xy – 30y=0

∴ y (6x-30) =0

y=0, x= 5

For x=5; From Equation 3x^{2}+3y^{2}-30x+72=0, we get y2-1=0

Y=±1

Hence (4,0) , (6,0) , (5,1) ,(5,-1) are the stationary points.

STEP 2] Now,`r=(del^2f)/(delx^2)=6x-30;`

`"S"=(del^2f)/(delxdely)=6y;`

`t=(del^2f)/(dely^2)=6x-30`

STEP 3] for (x, y) ≡ (4, 0), r = -6, s = 0, t = -6;

rt –s^{2}=(-6)(-6)-0=36>0 and r< 0.

This shows that the function is maximum at (4, 0)

∴ From Equation (1)

F_{max}=f (4, 0) =4^{3}+0-15(4^{2}) +0+72(4) =64 – 240 + 288

F_{max}=112

STEP 4] For (x,y)≡(6,0)

r=6, s=0, t=6

rt-s^{2}=36 but r=6>0

This shows that function is minimum at (6, 0).

From Equation (1),

F_{min}=f(6,0)=6^{3}+0-15(6)^{2}+0+72(6)=108.

STEP 5] For(x, y) ≡ (5, 1)

r=0, s=6, t=0;

(rt-s^{2})<0

This shows that at (5, 1) and (5,-1) function is neither maxima nor minima.

These points are saddle points.