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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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?

Similar Questions

$A$ function $y = f(x)$ is given by $x = \frac{1}{1 + t^2}$ and $y = \frac{1}{t(1 + t^2)}$ for all $t > 0$. Then $f$ is:

The function $f(x) = 2x^3 - 3x^2 + 90x + 174$ is increasing in the interval

Prove that the function $f$ given by $f(x)=x^{2}-x+1$ is neither strictly increasing nor decreasing on $(-1,1).$

In which of the following intervals is the function $f(x) = 2x^3 - 15x^2 + 36x + 1$ strictly decreasing?

For what values of $x$ does the function $f(x) = x^2 e^{-x}$ strictly increase?

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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?