Question

if a and b are nonzero constants, find the domain and all critical points of f(x)= (ax^2)/(x-b)
if a and b are nonzero constants, find the domain and all critical points of f(x)= (ax^2)/(x-b)
@Mathematics

Answer 1

\[\text{all real numbers,} x \neq b \]

Answer 2

use the quotient rule to find f'

Answer 3

\[f'(x)=\frac{(ax^2)'(x-b)-ax^2(x-b)'}{(x-b)^2}=\frac{2ax(x-b)-ax^2(1-0)}{(x-b)^2}\]
\[=\frac{2ax^2-2axb-ax^2}{(x-b)^2}=\frac{ax(2x-2b-x)}{(x-b)^2}=\frac{ax(x-2b)}{(x-b)^2}\]
when is the numerator 0?
when ax=0 or when x-2b=0
solve both for x and voolah critical numbers
also you can say x=b is a critical number since f' dne exist there