Question 133084
To find the min/max of {{{-3x^2+24x}}}, we need to find the vertex. However, we first need to find the axis of symmetry




To find the axis of symmetry, use this formula:


{{{x=-b/(2a)}}}


From the equation {{{y=-3x^2+24x}}} we can see that a=-3 and b=24


{{{x=(-24)/(2*-3)}}} Plug in b=24 and a=-3



{{{x=(-24)/-6}}} Multiply 2 and -3 to get -6




{{{x=4}}} Reduce



So the axis of symmetry is  {{{x=4}}}



So the x-coordinate of the vertex is {{{x=4}}}. Lets plug this into the equation to find the y-coordinate of the vertex.



Lets evaluate {{{f(4)}}}


{{{f(x)=-3x^2+24x}}} Start with the given polynomial



{{{f(4)=-3(4)^2+24(4)}}} Plug in {{{x=4}}}



{{{f(4)=-3(16)+24(4)}}} Raise 4 to the second power to get 16



{{{f(4)=-48+24(4)}}} Multiply 3 by 16 to get 48



{{{f(4)=-48+96}}} Multiply 24 by 4 to get 96



{{{f(4)=48}}} Now combine like terms



So the vertex is (4,48)



Now since the leading coefficient is -3, this means that the parabola opens down. So at the vertex there is a maximum. Also, this means that the max is {{{y=48}}} which occurs at {{{x=4}}}