) + = ( x – ) + > , in every interval of R Therefore, the function f is increasing on R . Example Prove that the function given by f ( x ) = cos x is (a) decreasing in ( , π ) (b) increasing in ( π , π ), and (c) neither increasing nor decreasing in ( , π ). Fig . Solution Note that f ′ ( x ) = – sin x (a) Since for each x ∈ ( , π ), sin x > , we have f ′ ( x ) < and so f is decreasing in ( , π ).
(b) Since for each x ∈ ( π , π ), sin x < , we have f ′ ( x ) > and so f is increasing in ( π , π ). (c) Clearly by (a) and (b) above, f is neither increasing nor decreasing in ( , π ). Example Find the intervals in which the function f given by f ( x ) = x – x + is (a) increasing (b) decreasing Solution We have f ( x ) = x – x + f ′ ( x ) = x – Therefore, f ′ ( x ) = gives x = . Now the point x = divides the real line into two disjoint intervals namely, (– ∞ , ) and ( , ∞ ) (Fig .
). In the interval (– ∞ , ), f ′ ( x ) = x – < . Therefore, f is decreasing in this interval. Also, in the interval ( , ∞ , x > ′ and so the function f is increasing in this interval.
Example Find the intervals in which the function f given by f ( x ) = x – x – x + is (a) increasing (b) decreasing. Solution We have f ( x ) =