How do you find the maximum value of #y = -2x^2 + 36x - 177#?

1 Answer
Aug 20, 2016

I got #-15#.


Since this function is a quadratic (#ax^2 + bx + c#), and since the second-degree coefficient is negative, this function has one maximum (look at the shape of any #f(x) = -|c|x^2#).

If you take the first derivative, #d/(dx)#, and set the result equal to #0#, you are finding the instantaneous slope at a maximum or minimum, and you now know that it will be a maximum.

#d/(dx)[-2x^2 + 36x - 177]#

#= -4x + 36#
(refer back to the Power Rule: #d/(dx)[x^n] = nx^(n-1)#.)

So, setting it equal to #0#:

#0 = -4x + 36#

#4x = 36#

#color(green)(x = 9)#

Now that you know what #x# value corresponds to the maximum value, the maximum value itself is the value of #f(x)#. Therefore, plug #x = 9# into #f(x)#:

#color(blue)(f(9)) = -2(9)^2 + 36(9) - 177#

#= -162 + 324 - 177#

#= -339 + 324#

#= color(blue)(-15)#

So, your maximum value is #f(9) = -15#, or the coordinates of your maximum is #color(blue)((9"," -15))#.