Question

Which of the following is not a decreasing function on the interval $\left(0,\frac{\pi }{2}\right)$?

• A. $\cos x$
• B. $\cos 2x$
• C. $\cos 3x$
• D. $cotx$

Answer 1

The correct option is C

$\cos 3x$

Explanation for the correct options:

Step 1:Find the derivative of the given function

If a function is decreasing in the inteval $\left(a,b\right)$, then $f\text{'}\left(x\right)<0$ in the interval $\left(a,b\right)$.

Let $f\left(x\right)=\cos 3x$

Then, $f\text{'}\left(x\right)=-3\sin 3x$.

We know, $0<x<\frac{\pi }{2}$

Multiply $3$

$0<3x<\frac{3\pi }{2}$

So, $3x\in \left(0,\pi \right)\cup \mathrm{\pi }\cup \left(\mathrm{\pi },\frac{3\mathrm{\pi }}{2}\right)$

Step 2: Finding the interval for decreasing and increasing.

$\sin \left(3x\right)>0$ for $3x\in \left(0,\pi \right)$

So, $-3\sin \left(3x\right)<0$ for $x\in \left(0,\frac{\pi }{3}\right)$

Again,

$\sin \left(3x\right)<0$ for $3x\in \left(\pi ,\frac{3\pi }{2}\right)$

$-3\sin \left(3x\right)>0$ for $x\in \left(\frac{\pi }{3},\frac{\pi }{2}\right)$

The $\cos 3x$ curve is decreasing from $\left(0,\frac{\pi }{3}\right)$ and increasing from $\left(\frac{\pi }{3},\frac{\pi }{2}\right)$.

Explanation for the incorrect options:

For option (A)

Let $f\left(x\right)=\cos x$

$f\text{'}\left(x\right)=-\sin \left(x\right)$

We know that $\sin x$ is increasing in the interval $\left(0,\frac{\pi }{2}\right)$

So, $-\sin x$ is decreasing in the interval $\left(0,\frac{\pi }{2}\right)$.

For option (B)

Let $f\left(x\right)=\cos 2x$

Then, $f\text{'}\left(x\right)=-2\sin 2x$.

We know, $0<x<\frac{\pi }{2}$

Multiply $2$

$0<2x<\pi$

$\sin \left(2x\right)>0$ for $2x\in \left(0,\pi \right)$

So, $-2\sin \left(2x\right)<0$ for $x\in \left(0,\frac{\pi }{3}\right)$

The $cos2x$ curve is decreasing from $\left(0,\frac{\pi }{2}\right)$.

For option (D)

Let, $f\left(x\right)=cotx$

So, $f\text{'}\left(x\right)=-\cos e{c}^{2}x$
We know $\cos ecx>0$ in the interval $\left(0,\frac{\mathrm{\pi }}{2}\right)$

So, $-\cos e{c}^{2}x<0$ in the interval $\left(0,\frac{\mathrm{\pi }}{2}\right)$.

The $cotx$ curve is decreasing from $\left(0,\frac{\pi }{2}\right)$.

Hence, $\cos 3x$ curve is decreasing from $\left(0,\frac{\pi }{3}\right)$ and increasing from $\left(\frac{\pi }{3},\frac{\pi }{2}\right)$.

Therefore, the correct option is option (C).