A manufacturer believes that the net profit P is related to the advertising expenditure x by P(x)=10+46x-(1/2)x^2 what advertising expenditure will maximize the profit

Question

anonymous · Answer

Take the derivative of P(x)
$$P'(x)=-x+46$$

To find the maximum, set the derivative equal to zero.
$$0=-x+46$$$$x=46$$

anonymous · Answer

Are you in calculus?  If not, there is also an algebraic way to do this.

nomi · Answer

algebra

anonymous · Answer

Oh, shoot, sorry.
Alright, we know that the x-coordinate of the vertex is located at 
$$x=-\frac{b}{2a}$$
So we need to figure out a, b, c before we can use that formula.
Let's put it in the form
$$0=ax^2+bx+c$$$$0=-\frac{1}{2}x+46x+10$$This means a=-1/2, b=46, c=10.

$$x=-\frac{b}{2a}$$$$x=-\frac{46}{-(2)(\frac{1}{2})}=(-46)(-1)=46$$$$x=46$$

nomi · Answer

nice awesome so you change the expression to quadratic right?

anonymous · Answer

I put it in general quadratic form.  It was a quadratic before, but the general form makes it easy to identify what a, b, and c are.