Question

Does the function `f(x)= -x^2+6x-1` have a minimum or maximum value?

Answer 1

The parabola will have a maximum value because the `x^2` term is negative.

Explanation:

1. Because the function has the general form `f(x)=color(blue)(A)x^2+color(purple)(B)x+color(red)(C)`, we know the graph will be a parabola.
2. The sign of the `x^2` term will tell us if the parabola opens up (like a `uu`) or down (like a `nn`):

If `A >0`, opens up (`uu`)

If `A<0`, opens down (`nn`)

In this case,
`f(x)=color(blue)(-(1))x^2color(purple)(+6)xcolor(red)(-1)`
`color(blue)(A = -1)` so the parabola will open "down" or `nn` which means the parabola will have a "peak" or maximum point.

graph{-x^2+6x-1 [-15, 15, -10, 10]}