Answer:
a= 3/2
b= 1
a*b= 3/2
Step-by-step explanation:
For getting this problem we need to think in terms of only one unknown. Lets start by looking at our equation:
2a + 3b=6
No, we need to see what happens to a*b. Lets get the value of one of these unknowns in terms of the other, this is, a in terms of b or b in terms of a. Here, I will get a in terms of b (if you like, do the other way and compare):
2a + 3b=6
Subtract 3b in both sides:
2a = 6 - 3b
Now divide both sides by 2 to get a alone:
a = (6 - 3b)/2
a = 3 - 3/2 b
Now lets try to multiply a by b, using this a we found above:
a*b = (3 - 3/2 b)*b
Using distributive:
a*b = 3b - 3/2 b^2
So, we have an expression for a*b that depends only on b. Notice that this expression is a parabola with negative coefficient on the square term, what makes it a negative parabola that MUST have a maximum value. So, we have an expression for a*b that can be maximized, so we can find the maximum of a*b by the derivative of the expression. Lets derive in b:
(3b - 3/2 b^2)' = (3b)' - (3/2 b^2)' = 3 - 3b
So, the derivative equal to 0 gives us the maximum:
3 - 3b = 0
Suming 3b in both sides:
3 = 3b
Dividing by 3:
1 = b
So, we maximize our expression a*b when b is equal to 1. Now, we can replace it on a = 3 - 3/2 b to find b:
a = 3 - 3/2(1) = 3 - 3/2 = 3/2
Thus, a is equal to 3/2 and the product a*b is maximized when:
a = 3/2
b = 1
And the product is a*b= (3/2) * 1 = 3/2
