Question 1200637
.
The number of tickets sold each day for an upcoming performance of​ Handel's Messiah 
is given by N=−0.5x2+13x+12​, where x is the number of days since the concert 
was first announced. When will daily ticket sales peak and how many tickets 
will be sold that​ day?
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<pre>
The given function N(x) = −0.5x^2 + 13x + 12  is a parabola opened downward.
It has a maximum.


Its optimal value of x can be found using the formula  {{{x[max]}}} = {{{-b/(2a)}}},

where "a" is the coefficient at x^2 and "b" is the coefficient at x.


In our case, a= -0.5,  b= 13,  so  {{{x[max]}}} = {{{-13/(2*(-0.5))}}} = {{{13/1}}} = 13.


To find the maximum value of the function, substitute x= 13 into the formula N(x)

    N(13) = -0.5*13^2 + 13*13 + 12 = 96.5.


At this point, turn on your attention: the number of tickets must be integer number, 
but we got a non-integer value.


At this point, turn on your common sense and consider another integer argument in vicinity of 13,
which, possible, will give you integer number of N(x).


Try x= 12. You will have  

    N(12) = -0.5*12^2 + 13*12 + 12 = 96.


It is just much better and more reasonable.


So, the answer is: the daily ticket sales peak is 96 tickets on 12-th day.


Notice, that the other possible answer is 96 tickets on 14-th day, too.
</pre>

Solved.


So, the solution is a standard algebraic procedure, but we should turn on our common sense 
and make the necessary corrections in order for the solution be meaningful.



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On finding the maximum/minimum of a quadratic function see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-to-find-the-vertex-of-a-quadratic-function.lesson>HOW TO complete the square to find the vertex of a parabola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-finding-the-vertex-of-a-parabola.lesson>Briefly on finding the vertex of a parabola</A>



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