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

Question

Cameron Alexander · Accepted Answer

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))#.

Samantha Adkison · Answer

undefined To find the maximum value of the quadratic function y = -2x^2 + 36x - 177, you can use the formula for the x-coordinate of the vertex of a quadratic function, which is given by x = -b/(2a), where a is the coefficient of the x^2 term and b is the coefficient of the x term. Once you find the x-coordinate of the vertex, substitute it back into the original function to find the corresponding y-coordinate. This y-coordinate will be the maximum value of the function.