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

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(D) For statement $S$: In the interval $\left( \frac{\pi}{2}, \pi \right)$,the derivative of $\sin x$ is $\cos x$,which is negative. Thus,$\sin x$ is

Vedclass · Accepted Answer

$S$ is true and $R$ is false.

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(D) For statement $S$: In the interval $\left( \frac{\pi}{2}, \pi \right)$,the derivative of $\sin x$ is $\cos x$,which is negative. Thus,$\sin x$ is decreasing. The derivative of $\cos x$ is $-\sin x$,which is also negative in this interval. Thus,$\cos x$ is also decreasing. Therefore,$S$ is true.For statement $R$: Consider the function $f(x) = \frac{1}{x}$ for $x \in (1, 2)$. Here,$f'(x) = -\frac{1}{x^2}$,which is negative,so $f(x)$ is decreasing. However,$f''(x) = \frac{2}{x^3}$,which is positive for $x \in (1, 2)$,meaning $f'(x)$ is an increasing function. Thus,the statement $R$ is false.

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