Grade 12Integral Calculus
the function f(x) = jx??2j
x2 is unde ned at x = 0 and jx??2j changes sign at x = 2 hence we divide the domain
of f(x) in to three intervals (??1; 0) (0; 2) (2;1)
f(x) =
8><
>:
2??x
x2 x 2 (??1; 0)
2??x
x2 x 2 (0; 2)
x??2
x2 x 2 (2;1)
nd the derivative for each of these intervals f0(x) =
8><
>:
x2??4x
x4 x 2 (??1; 0)
x2??4x
x4 x 2 (0; 2)
4x??x2
x4 x 2 (2;1)
.
As x4 is always positive determine the sign of numerator in each and hence nd the intervals of mono-
tonicity.
Increasing in (??1; 0) [ (4;1)
Decreasing in (0; 2) [ (2; 4).
