Question
The function $$f\left( x \right) = 1 + x\left( {\sin \,x} \right)\left[ {\cos \,x} \right],\,0 < x \leqslant \frac{\pi }{2}$$ (where [.] is G.I.F.)
A.
is continuous on $$\left( {0,\,\frac{\pi }{2}} \right)$$
B.
is strictly increasing in $$\left( {0,\,\frac{\pi }{2}} \right)$$
C.
is strictly decreasing in $$\left( {0,\,\frac{\pi }{2}} \right)$$
D.
has global maximum value $$2$$
Answer :
is continuous on $$\left( {0,\,\frac{\pi }{2}} \right)$$
Solution :
For $$0 < x \leqslant \frac{\pi }{2}\,;\,\left[ {\cos \,x} \right] = 0$$
Hence, $$f\left( x \right) = 1$$ for all $$\left( {0,\,\frac{\pi }{2}} \right]$$
Trivially $$f\left( x \right)$$ is continuous on $$\left( {0,\,\frac{\pi }{2}} \right)$$
This function is neither strictly increasing nor strictly decreasing and its global maximum is $$1.$$