So, what is your problem? What don't you understand? You got it correctly [except "0 = b(1) -2". That should have been 0 = b -2(1)], so ....?

You used derivatives--yes, the derivative is correct, and yes, at the "apex" or vertex, the slope of the tangent line, or the 1st derivative, is zero.

So, what is wrong?

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"...but the way I am used to use for finding apexes don't work with the formula y=a+bx-x^2."

And what is that way?

The "x = -b/2a" way?

y = a +bx -x^2

y' = b -2x

y' = 0

0 = b -2x

2x = b

x = b/2 --------the x-coordinate of the apex/vertex of y = a +bx -x^2.

Is that wrong?

Is this the reason why it does not work with y = a +bx -x^2 ?

y = a +bx -x^2

Rewriting that,

y = -x^2 +bx +a ----(1)

At the vertex,

x = "-b/2a"

Here, in (1), "b" is +b; and "a" is -1, so,

x = -(+b)/[2(-1)]

x = -b/-2

x = b/2 ----------is that the same as the x = b/2 above?