question_answer

Consider the following statements S and R S : Both \[\sin x\] and cosx are decreasing functions in \[\left( \frac{\pi }{2},\pi  \right)\] R : If a differentiable function decreases in (a, b)  then its derivative also decreases in (a, b). Which of the following is true [IIT Screening 2000]

A)            Both S and R are wrong   

B)            Both S and R are correct but R is not the correct explanation for S

C)            S is correct and R is the correct explanation for S     

D)            S is correct and R is wrong

Correct Answer: D

Solution :

           From the trend of value of \[\sin x\] and \[\cos x\] we know \[\sin x\] and \[\cos x\] decrease in \[\frac{\pi }{2}<x<\pi \]. So, the statement S is correct.                 The statement R is incorrect which is clear from graph. Clearly \[f(x)\] is differentiable in   (a, b).            Also,\[a<{{x}_{1}}<{{x}_{2}}<b\].            But \[{f}'({{x}_{1}})=\tan {{\varphi }_{1}}<\tan {{\varphi }_{2}}={f}'({{x}_{2}}).\]