Find the maximum and minimum values of fx y 5c y on the el

Find the maximum and minimum values of f(x, y) = 5c + y on the ellipse x^2 + 9y^2 = 1 maximum value: minimum value You have attempted this problem 5 times. Your overall recorded score is 0%. You have unlimited attempts remaining.

Solution

f(x,y)=5x+ y

let g(x,y)= x²+9y²-1

by the method of lagrange meltipliers

f = g

<5,1> = <2x ,18y>

5 =2x , 1 =18y

=5/(2x) ,=1/(18y)

5/(2x) =1/(18y)

2x =90y

x =45y

x²+9y²=1

(45y)²+9y²=1

=>2034y²=1

=>y²=1/2034

=> y =-1/2034 ,y =1/2034

x =45y

=> x =-45/2034 ,x =45/2034

(x,y)=(-45/2034 , -1/2034 ) ,(x,y)=(45/2034 ,1/2034 )

f(-45/2034 , -1/2034 ) =5(-45/2034) +( -1/2034 ) =-226/2034

f(45/2034 , 1/2034 ) =5(45/2034) +( 1/2034 ) =226/2034

maximum value =226/2034 =(226)/3

minimum value =-226/2034 =(-226)/3