find the minimum value of 2x+y when

Question

xy=8 ; x,y ∈R₊

Answer 1

If y=8/x then 2x+y=2x+8/x and as you can see when x is small and >0 this function is large and when x is really big and >0 this function is large again so somewhere in between this function has a minimum value. To find it you differentiate with respect to x and set the resulting function equal to zero.

 

d/dx(2x+8/x)=2-8/x²=0 solving for x gives 4/x²=1 or x=2. So at x=2 (2*2+8/2)=8 is the minimum value

 

 

Jim

Answer 2

x and y are two positive numbers that when multiplied together, you get a result of 8.

 

Here are your ordered pairs:

 

(1 , 8) , (2, 4) , (4 , 2) ,  and (8 , 1)

 

 

Now plug in these value of ordered pairs into the expression 2x + y.

 

2(1) + 8 = 2 + 8 = 10

2(2) + 4 = 4 + 4 = 8

2(4) + 2 = 8 + 2 = 10

2(8) + 1 = 16 + 1 = 17

 

 

The minimum result here is 8.