Is f(x)=e^x/sqrt(x^2-x) increasing or decreasing at x=3?

1 Answer
Jun 13, 2018

Calculate the derivative via the quotient rule:
f'(x)=(e^xsqrt(x^2-x)-e^x(2x-1)/(2sqrt(x^2-x)))/(x^2-x)
=e^x(x^2-2x+1/2)/(x^2-x)^(3/2)

If this is positive, then f is increasing.

f'(3)=e^3(9-6+1/2)/(9-3)^(3/2)=7/2e^3 6^(3/2)>0, because each of its component multiplicative parts is >0. So f(x) is increasing at x=3.

For a sanity check, compare the graph of the function:
graph{(e^x)/sqrt(x^2-x) [-9.26, 14.7, 2, 13.99]}