Find the maximum and minimum values of the function f(x,y) = 2x – 7y subject to the constraint x² + y² = 9. In the answer mention

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1. Answer:

   Solution :

   Given f(x,y)=x3-4xy+2y2 differentiating equation(i),we have

   fx= df/dx = 3 x 2-4y

   fy= df/dy = -4x+4y

   and r = f xx = 6 x

   s= f xy = -4

   t=f yy=4

   Now for maxima or minima,we must have fx=0,fy=0,we have

   3×2-4y=0 (ii)

   and -4x+4y=0 (iii)

   Solving equations (ii) and (iii) , we get

   3×2-4x=0 or x(3x-4)=0 or x=0, 4/3

   Now from equation (iii), we have

   when x=0 => y=0 and when x=(4/3) => y=(4/3)

   thus the reuired stationary points are (0,0) and (4/3,4/3)

   At point (4/3,4/3)

   r=6 X (4/3) =8

   s=-4

   t=4

   rt-s2=8 X 4 – ( -4)2=32-16=16 > 0 and r > 0

   Hence ,f(x,y) has a minima at (4/3,4/3)

   At point (0,0)

   r=6 X 0 =0 s=-4

   t=4

   rt-s2 = 0-(-4)2 = -16 = -ve

   and hence there is neither maxima nor minima at (0,0)

   Minimum value of f(x,y) =[x3 – 4xy = 2y2] x = 4/3 , y = 4/3

   =(4/3)3-4.(4/3).(4/3)+2.(4/3)2

   =(64/27)-(64/9)+(32/9)

   =(64-192+96)/27

   =-(32/27) Ans

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