Math 31 calculus

Question

Find the interval on which the function f(x)=x/(x²+1) is increasing.

Answer 1

f(x) = x(x²+1)^-1

df(x)/dx = (x²+1)^-1- x·2x·(x²+1)^-2

df(x)/dx = (x²+1)^-1- 2x²·(x²+1)^-2

0 = (x²+1)^-1- 2x²·(x²+1)^-2

2x²/(x²+1)² = 1/(x²+1)

2x² = x² + 1

x² = 1

x = ±1

Take the second derivative and plug in x = 1 and x= -1 to see if each extrema is a mx or min. It's a maximum if f ''(x) < 0. It's a minimum if f ''(x) > 0. The increasing interval is between the minimum and maximum.