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# 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.

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#### 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
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