What are the critical points of f(x) =(x^2-4x+5)/(x-3)f(x)=x2−4x+5x−3?

1 Answer
Jim H
Feb 4, 2016

The critical numbers are 11 and 77.

Explanation:

f'(x) = ((2x-4)(x-3)-(x^2-4x+5)(1))/(x-3)^2

= (x^2-6x+7)/(x-3)^2

f' fails to exists only at x=3 which is not in the domain of f,

and f'(x) = 0 at x=1 and at x=7, both of which are in the domain of f.

So, the critical numbers for f are 1 and 7.

There appears to be some variability in the use of the term "critical point".
I use it to mean a point in the domain of a function at which the derivative is 0 or fails to exist.
Some people seem to use it to mean a point on the graph of a function at which the derivative is 0 or fails to exist.

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