What are the points of inflection, if any, of #f(x)=(x-3)/(x^2-4x+5) #?
Use the quotient rule to find
You should get
# = (-x^2+6x-7)/(x^2-4x+5)^2#
# = ((-2x+6)(x^2-4x+5)-(-x^2+6x-7)(2(2x-4)))/(x^2-4x+5)^3#
# = (2(x^3-9x^2+21x-13))/(x^2-4x+5)^3#
To see where the concavity changes analyze the sign of
The denominator is never
Find the remaining zeros by the quadratic formula.
The zeros of the numerator are
None of these zeros have even multiplicity, so the sign of
To find the corresponding
Here is the graph of