Determine the intervals on which the function is increasing

Determine the intervals on which the function is increasing, decreasing, or constant. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

x2 + x +1 x+ 1 f(x) = _ increasing (-oo, - 2).(0,00) decreasing constant DNE 5 2,-3)

Solution

f(x) = (x² +x + 1)/(x + 1)

f \'(x) = [(2x +1)(x +1) - (x² + x +1)(1)]/(x +1)²          since (u/v)\' = [u\'v - uv\']/v²

==> f \'(x) = [2x² + 2x + x +1 - x² - x -1]/(x +1)²      

==> f \'(x) = [x² + 2x]/(x +1)²

a) function is increasing ==> f \'(x) > 0

==> [x² + 2x]/(x +1)² > 0          ; (x +1)² is always greater than zero

==> x² + 2x > 0

==> x(x +2) > 0

==> x > 0 or x < -2

==> Interval of increase ==> (- , -2) U (0 , )

function is decreasing ==> f \'(x) < 0

==> [x² + 2x]/(x +1)² < 0 ; (x +1)² is always greater than zero

==> x² + 2x <0

==> x(x +2) < 0

==> - 2 < x < 0

but the function is undefined at x = -1

==> interval of decrease = (-2 , -1) U (-1 , 0)