Consider the following statements S and R:S: | vedclass

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(D) Statement $S$: In the interval $\left( \frac{\pi}{2}, \pi \right)$,$\sin x$ decreases from $1$ to $0$,and $\cos x$ decreases from $0$ to $-1$. The

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$S$ is correct and $R$ is wrong.

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(D) Statement $S$: In the interval $\left( \frac{\pi}{2}, \pi \right)$,$\sin x$ decreases from $1$ to $0$,and $\cos x$ decreases from $0$ to $-1$. Therefore,both functions are decreasing in this interval. Thus,statement $S$ is correct.Statement $R$: If a function $f(x)$ is decreasing in $(a, b)$,it implies $f'(x) \le 0$. It does not necessarily imply that $f'(x)$ is a decreasing function. For example,consider $f(x) = -x^3$ on $(-1, 1)$. Here $f'(x) = -3x^2$,which is not a decreasing function on $(-1, 1)$. The provided graph also illustrates a case where a function is decreasing,but its slope (derivative) is increasing. Thus,statement $R$ is incorrect.Conclusion: $S$ is correct and $R$ is wrong. The correct option is $D$.

Consider the following statements $S$ and $R$:
$S$: Both $\sin x$ and $\cos x$ 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?

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Consider the following statements $S$ and $R$:$S$: Both $\sin x$ and $\cos x$ 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?