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The positive numbers 
x and 
a-x have a sum of 
a. What is 
x in terms of 
a if the product 
x*(a-x) is a maximum?
Choose 1 answer:
(A) 
(a)/(2)
(B) 
sqrt((a)/(2))
(C) 
sqrta
(D) 
a
E There is no 
x that would produce a maximum product

The positive numbers x x and ax a-x have a sum of a a . What is x x in terms of a a if the product x(ax) x \cdot(a-x) is a maximum?\newlineChoose 11 answer:\newline(A) a2 \frac{a}{2} \newline(B) a2 \sqrt{\frac{a}{2}} \newline(C) a \sqrt{a} \newline(D) a a \newline(E) There is no x x that would produce a maximum product

Full solution

Q. The positive numbers x x and ax a-x have a sum of a a . What is x x in terms of a a if the product x(ax) x \cdot(a-x) is a maximum?\newlineChoose 11 answer:\newline(A) a2 \frac{a}{2} \newline(B) a2 \sqrt{\frac{a}{2}} \newline(C) a \sqrt{a} \newline(D) a a \newline(E) There is no x x that would produce a maximum product
  1. Equation Simplification: We know that x+(ax)=ax + (a - x) = a. This simplifies to x+ax=ax + a - x = a, which means xx cancels out and we're left with a=aa = a, which is true but doesn't help us find xx.
  2. Maximizing Product: To maximize the product x(ax)x(a-x), we can use the fact that for a fixed sum, the product of two numbers is maximized when the numbers are equal. So, we set x=axx = a - x.
  3. Solving for x: Solving for x gives us x=axx = a - x, which simplifies to 2x=a2x = a.
  4. Isolating xx: Divide both sides by 22 to isolate xx, we get x=a2x = \frac{a}{2}.

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