Question 319919
You can find the max. value by putting the equation into vertex form,{{{a(x-h)^2+k}}}, by completing the square.
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{{{ P(x) =-0.0004x^2 +12x-1500}}}
{{{P(x)=-0.0004(x^2-30000x)-1500}}}
{{{P(x)=-0.0004(x^2-30000x+225000000)-1500+90000}}}
{{{P(x)=-0.0004(x-15000)^2+88500}}}
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The minimum occurs at the vertex.
a){{{Pmax=88500}}}
b){{{x}}} at {{{Pmax}}}:{{{x=15000}}}