Similarly,
if f¢(x) < 0 for x in an interval excluding the end points and f is continuous in the
interval, then f is decreasing Example 8 Show that the function  f  given by
f (x) = x3 – 3x2 + 4x, x ∈ R
is increasing on R Solution Note that
f ′(x) = 3x2 – 6x + 4
= 3(x2 – 2x + 1) + 1
= 3(x – 1)2 + 1 > 0, in every interval of R
Therefore, the function  f is increasing on R Example 9 Prove that the function given by f (x) = cos x is
(a)
decreasing in (0, π)
(b)
increasing in (π, 2π), and
(c)
neither increasing nor decreasing in (0, 2π)