Question

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

Answer 1

I got `-15`.

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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))`.