Question 173539
I always think of the maximum as being exactly between
the 2 roots of a parabola (where it crosses the x-axis
The quadratic formula finds the roots
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
There is a (+) and a (-) answer, and thoses are the roots
If you rewrite the formula like this:
{{{x = -b/2a +- sqrt( b^2-4*a*c )/(2*a) }}}
you can see that {{{-b/2a}}} is in the middle and you add
the 2nd term to get the larger root and subtract the 2nd
term to get the smaller root.
So, you just have to find {{{-b/2a}}} to find the maximum
The general formula for finding roots is
{{{ax^2 + bx + c = 0}}}
The equation in the problem is ({{{d}}} replaces {{{x}}})
{{{I(d) = -120d^2 + 14400d + 100}}}
{{{a = -120}}}
{{{b = 14400}}}
{{{c = 100}}}
{{{-b/2a = -14400/(-240)}}}
{{{-b/2a = 60}}}
This says {{{d = 60}}} when {{{I(d)}}} is a maximum
$60 /widget should be charged to maximize income
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The problem wants to know what that income is
{{{I(d) = -120d^2 + 14400d + 100}}}
{{{I(60) = -120*60^2 + 14400*60 + 100}}}
{{{I(60) = -120*3600 + 864000 + 100}}}
{{{I(60) = -432000 + 864000 + 100}}}
{{{I(60) = 432100}}}
The maximum income is $432,100
You can check the answers by making {{{d}}} a little
bit less, say {{{59.9}}} and finding {{{I(d)}}}
and a little bit more, say {{{60.1}}} and finding
{{{I(d)}}} In both cases, {{{I(d)}}} should be 
less than {{{432100}}}