# Thread: Maximum value of combined function

1. ## Maximum value of combined function

Hi,

I hope someone can help. I'm currently working on 10c:

4. ## Re: Maximum value of combined function

Let's multiply out:

$R = 500,000+1250x-750x^2 = -750\left( x^2-\dfrac{5}{3}x-\dfrac{2000}{3} \right)$

R is revenue.

Now, let's complete the square:

$R = -750\left( x^2-\dfrac{5}{3}x+\dfrac{25}{36}-\dfrac{25}{36}-\dfrac{2000}{3} \right) = -750\left(x-\dfrac{5}{6}\right)^2+\dfrac{3003125}{6}$

This has a maximum at $x=\dfrac{5}{6}$ (because it is a parabola).

So, it is maximized when the price is $25+x = 25+\dfrac{5}{6} \approx \$25.83\$

5. ## Re: Maximum value of combined function

The formula that skeeter used, "x= -b/2a" can be derived by "completing the square", as SlipEternal did, in the general quadratic equation. $ax^2+ bx+ c= a(x^2+ (b/a)x)+ c$. [tex]a(x^2+ (b/a)x+ b^2/4a^2- b^2/4a^2)+ c= a(x^2+ (b/a)x+ b^2/4a^2)- b^2/4a+ c= a(x+ b/2a)^2+ (c- b^2/4a).

If x= -b/2a then the quantity squared is 0 and the value of the function is c- b^2/4a.

If x is any other number then, because a square is never negative, if a is negative, this is c- b^2/4a minus something so less than c- b^2/4a and c- b^2/4a is the maximum of the function. If a is positive, this is c- b^2/4a plus something so larger than c- b^2/4a and c- b^2/4a is the minimum value of the function.

6. ## Re: Maximum value of combined function

Great! This makes sense. Thanks for everyone's help.