Which of the following functions are strictly | vedclass

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

Question

(A, B) Let $f_{1}(x) = \cos x$. Then $f_{1}^{\prime}(x) = -\sin x$. In the interval $\left(0, \frac{\pi}{2}\right)$,$\sin x > 0$,so $f_{1}^{\prime}(x)

Vedclass · Accepted Answer

$\cos 2x$

Vedclass · Answer

(A, B) Let $f_{1}(x) = \cos x$. Then $f_{1}^{\prime}(x) = -\sin x$. In the interval $\left(0, \frac{\pi}{2}ight)$,$\sin x > 0$,so $f_{1}^{\prime}(x) = -\sin x Thus,$f_{1}(x) = \cos x$ is strictly decreasing on $\left(0, \frac{\pi}{2}ight)$. Let $f_{2}(x) = \cos 2x$. Then $f_{2}^{\prime}(x) = -2 \sin 2x$. For $0  0$. Thus,$f_{2}^{\prime}(x) = -2 \sin 2x Therefore,$f_{2}(x) = \cos 2x$ is strictly decreasing on $\left(0, \frac{\pi}{2}ight)$. Let $f_{3}(x) = \cos 3x$. Then $f_{3}^{\prime}(x) = -3 \sin 3x$. For $x \in \left(0, \frac{\pi}{2}ight)$,$3x \in (0, \frac{3\pi}{2})$. In $\left(0, \frac{\pi}{3}ight)$,$\sin 3x > 0$,so $f_{3}^{\prime}(x) In $\left(\frac{\pi}{3}, \frac{\pi}{2}ight)$,$\sin 3x  0$. Thus,$f_{3}(x)$ is not strictly decreasing on $\left(0, \frac{\pi}{2}ight)$. Let $f_{4}(x) = 	an x$. Then $f_{4}^{\prime}(x) = \sec^{2} x > 0$ for all $x \in \left(0, \frac{\pi}{2}ight)$. Thus,$f_{4}(x)$ is strictly increasing on $\left(0, \frac{\pi}{2}ight)$. Conclusion: Both $\cos x$ and $\cos 2x$ are strictly decreasing on $\left(0, \frac{\pi}{2}ight)$.

Which of the following functions are strictly decreasing on $\left(0, \frac{\pi}{2}\right)?$

Similar Questions

If $x \in (0, 1)$,then what type of function is $f(x) = x^{100} + \sin x - 1$?

For which value of $x$ is the function $f(x) = x^2 - 2x$ decreasing?

Let $y = x^2 e^{-x}$. The interval in which $y$ increases with respect to $x$ is:

The function,$f(x)=(3x-7)x^{2/3}, x \in R,$ is increasing for all $x$ lying in

For a real number $a$,if a real-valued function $f(x) = 4x^3 + ax^2 + 3x - 2$ is monotonic in its domain,then the range of $a$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module