Question

Assertion :Consider the following statements S and R.
Both $\sin x$ and $\cos x$ are decreasing function in $(\frac{\pi }{2},\pi ).$ Reason: If a differentiable function decreases in (a, b) then its derivative also decreases in (a, b).

• A. Assertion and Reason are correct and reason is the correct explanation of the assertion
• B. Assertion and Reason are correct and reason is NOT the correct explanation of the assertion
• C. Assertion is correct and Reason is incorrect
• D. Assertion is incorrect and reason is correct

Answer 1

The correct option is C Assertion is correct and Reason is incorrect

Let $f\left(x\right)=cos\left(x\right)$.
Then $f^{\prime }\left(x\right)=-sin\left(x\right)$

Now for monotonically decreasing
$-sin\left(x\right)<0$
Or $sin\left(x\right)>0$
Or $xϵ(0,\pi )$.

Hence $cos\left(x\right)$ is decreasing in $(0,\pi )$.
Now consider $f\left(x\right)=sin\left(x\right)$.
$f^{\prime }\left(x\right)=cos\left(x\right)$.

For monotonically decreasing
$cos\left(x\right)<0$
Or $xϵ(\frac{\pi }{2},\frac{3\pi }{2})$.

Hence $sin\left(x\right)$ is decreasing in the interval of $(\frac{\pi }{2},\frac{3\pi }{2})$.

Thus in the interval of $(\frac{\pi }{2},\pi )$ both $cos\left(x\right)$ and $sin\left(x\right)$ are decreasing.
Now again consider $f\left(x\right)=cos\left(x\right)$.
$f^{\prime }\left(x\right)=-sin\left(x\right)$.

Now for monotonically decreasing
$-sin\left(x\right)<0$

Or $sin\left(x\right)>0$
Hence even though $f\left(x\right)$ is decreasing in the interval, its derivative is increasing in that interval. Hence reason is not true.