Question

Consider the following statments in S and R
S : Both sin x and cos x are decreasing functions in the interval $\left(\frac{\pi }{2},\pi \right)$
R : If a differentiable function decreases in an interval (a, b), then its derivative also decreases in (a, b).
Which of the following is true ?

• A. Both S and R are wrong
• B. Both S and R are correct, but R is not the correct explanation of S
• C. S is correct and R is the correct explanation for S
• D. S is correct and R is wrong

Answer 1

Answer: (D)

From graph it is clear that both sin x and cos x in the interval $(\pi /2,\pi )$ are decreasing function.
$\therefore$ S is correct.
To disprove R let us consider the counter example :
f (x) = sin x on $(0,\pi /2)$
so that $f^{\prime }(x)=\cos x$
Again from graph it is clear that f (x) is increasing on $(0,\pi /2)$ but $f^{\prime }(x)$ is decreasing on $(0,\pi /2)$
$\therefore$ R is wrong.