Question 123462
Maximize {{{P(x)=-4x^2+16x-7}}}


Two ways to solve this.


Without calculus:


This function describes a convex down parabola, so the maximum point is at the vertex.  The x-coordinate of the vertex of parabola {{{f(x)=ax^2+bx+c}}} is at {{{x=-b/2a}}}


{{{x[max]=-16/2(-4)=2}}}


The value of the function at {{{x=2}}} is the maximum profit.


{{{f(2)=-4(2)^2+16(2)-7=-16+32-y=9}}}


So the profit is a maximum $9,000 when 2 units are sold.


Calculus:
A local minimum or maximum of a function is at the point where the first derivitive is equal to zero


{{{dy/dx=-8x+16}}}

{{{-8x+16=0}}}
{{{-8x=-16}}}
{{{x=2}}}


If the second derivitive evaluated at this value is < 0, the point is a maximum.


{{{d^2(y)/dx^2=-8}}}


Therefore the point is a maximum.


Find the function value at the maximum point just like before.