Question

The function f(x,y) = xy +
(a3÷x)+(a3÷y)at (a,a)has
​

Answer 1

Step-by-step explanation:

f(x,y)= xy + a^3*(1/x+1/y)

first of all, we will make b=a^3 to not write a^3 too much

f(x,y)= xy + b*(1/x+1/y)

df(x,y)= d[xy + b*(1/x+1/y)]

df(x,y)= d[xy] +d[ b*(1/x+1/y)]

df(x,y)= d[xy] +b*d[ (1/x+1/y)]

df(x,y)= xdy+ydx +b*[ (-1/x^2)dx +(-1/y^2)dy]

df(x,y)= xdy+ydx +[ (-b/x^2)dx +(-b/y^2)dy]

df(x,y)= [y-(b/x^2)]dx +[x-(b/y^2)]dy

maximum or minimum for x

→ df(x,y)/dx=0

→ y-(b/x^2)=0

→ y=b/x^2

→ y*x^2=b ………………1

maximum or minimum for y

→ df(x,y)/dy=0

→ x-(b/y^2)=0

→ x=b/y^2

→ x*y^2=b ………………2