X-coordinates of critical points are independent of m and n

Question

I was given a function $f(x)=a(-3x+2)^7-b,$ where a and b are constants that do not equal zero. I am supposed to Show that the x-coordinate(s) of the location(s) of any critical points are independent of a and b. I took the first derivative and second derivative already. I am stuck on how to show this. Thank you!

Answer 1

$f'(x) = -3a$ is very dependent upon $a$ and independent of $b$.
There are no critical points, points x for which $f'(x) = 0$, as $a$ is nonzero.
It makes no difference if $b$ is zero.
So as there are no critical points, their location,
namely nowhere, is independent of any nonzero $a$.

By the power of the lucky $7$,
$f'(x) = -21ax(-3x + 2)^6 = 0 $ which for nonzero $a$ is equivalent to $x(-3x + 2)^6 = 0. $

So the critical points are independent of $b$ and nonzero $a$.
Why second derivatives?
Are you looking for local minimums/maximums, inflection points?