Advertisement Remove all ads

Find Maximum and Minimum Values of X3 +3xy2 -15x2-15y2+72x. - Applied Mathematics 1

Sum

Find maximum and minimum values of x3 +3xy2 -15x2-15y2+72x.

Advertisement Remove all ads

Solution 1

Given f(x) = x3 +3xy2 -15x2-15y2+72x… (1)
STEP 1] for maxima, minima,`(delf)/(delx)=0;(delf)/(dely)=0`
3x2+3y2-30x+72=0 and 6xy – 30y=0
∴ y (6x-30) =0
y=0, x= 5
For x=5; From Equation 3x2+3y2-30x+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 –s2=(-6)(-6)-0=36>0 and r< 0.
This shows that the function is maximum at (4, 0)
∴ From Equation (1)
Fmax=f (4, 0) =43+0-15(42) +0+72(4) =64 – 240 + 288
Fmax=112
STEP 4] For (x,y)≡(6,0)
r=6, s=0, t=6
rt-s2=36 but r=6>0
This shows that function is minimum at (6, 0).
From Equation (1),
Fmin=f(6,0)=63+0-15(6)2+0+72(6)=108.
STEP 5] For(x, y) ≡ (5, 1)
r=0, s=6, t=0;
(rt-s2)<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) = x3 +3xy2 -15x2-15y2+72x… (1)
STEP 1] for maxima, minima,`(delf)/(delx)=0;(delf)/(dely)=0`
3x2+3y2-30x+72=0 and 6xy – 30y=0
∴ y (6x-30) =0
y=0, x= 5
For x=5; From Equation 3x2+3y2-30x+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 –s2=(-6)(-6)-0=36>0 and r< 0.
This shows that the function is maximum at (4, 0)
∴ From Equation (1)
Fmax=f (4, 0) =43+0-15(42) +0+72(4) =64 – 240 + 288
Fmax=112
STEP 4] For (x,y)≡(6,0)
r=6, s=0, t=6
rt-s2=36 but r=6>0
This shows that function is minimum at (6, 0).
From Equation (1),
Fmin=f(6,0)=63+0-15(6)2+0+72(6)=108.
STEP 5] For(x, y) ≡ (5, 1)
r=0, s=6, t=0;
(rt-s2)<0
This shows that at (5, 1) and (5,-1) function is neither maxima nor minima.
These points are saddle points.

Concept: Maxima and Minima of a Function of Two Independent Variables
  Is there an error in this question or solution?
Advertisement Remove all ads
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×