Question

Which of the following functions is decreasing in $\left(0,\frac{\mathrm{\pi }}{2}\right)$?
(a) sin 2x (b) tan x (c) cos x (d) cos 3x

Answer 1

Let f(x) = sin2x

$\therefore f\text{'}\left(x\right)=2\cos 2x$

$0<x<\frac{\mathrm{\pi }}{2}$ (Given)

$\Rightarrow 0<2x<\mathrm{\pi }$

Now, cos2x > 0 when $0<2x<\frac{\mathrm{\pi }}{2}$ and cos2x < 0 when $\frac{\mathrm{\pi }}{2}<2x<\mathrm{\pi }$.

$\Rightarrow f\text{'}\left(x\right)>0$ when $0<x<\frac{\mathrm{\pi }}{4}$ and $f\text{'}\left(x\right)<0$ when $\frac{\mathrm{\pi }}{4}<x<\frac{\mathrm{\pi }}{2}$

⇒ f(x) is increasing when $0<x<\frac{\mathrm{\pi }}{4}$ and f(x) is decreasing when $\frac{\mathrm{\pi }}{4}<x<\frac{\mathrm{\pi }}{2}$

Thus, f(x) = sin2x is both increasing and decreasing in the interval $\left(0,\frac{\mathrm{\pi }}{2}\right)$.


Let g(x) = tanx

$\therefore g\text{'}\left(x\right)={\sec }^{2}x$

Now, sec²x > 0 when $0<x<\frac{\mathrm{\pi }}{2}$

$\Rightarrow g\text{'}\left(x\right)>0$ when $0<x<\frac{\mathrm{\pi }}{2}$

⇒ g(x) = tanx is increasing when $0<x<\frac{\mathrm{\pi }}{2}$


Let h(x) = cosx

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

Now, sinx > 0 when $0<x<\frac{\mathrm{\pi }}{2}$

$\Rightarrow h\text{'}\left(x\right)<0$ when $0<x<\frac{\mathrm{\pi }}{2}$

⇒ h(x) = cosx is decreasing when $0<x<\frac{\mathrm{\pi }}{2}$


Let p(x) = cos3x

$\therefore p\text{'}\left(x\right)=-3\sin 3x$

$0<x<\frac{\mathrm{\pi }}{2}$ (Given)

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

Now, sin3x > 0 when $0<3x<\mathrm{\pi }$ and sin3x < 0 when $\mathrm{\pi }<3x<\frac{3\mathrm{\pi }}{2}$.

$\Rightarrow p\text{'}\left(x\right)<0$ when $0<x<\frac{\mathrm{\pi }}{3}$ and $p\text{'}\left(x\right)>0$ when $\frac{\mathrm{\pi }}{3}<x<\frac{\mathrm{\pi }}{2}$

⇒ p(x) is decreasing when $0<x<\frac{\mathrm{\pi }}{3}$ and p(x) is increasing when $\frac{\mathrm{\pi }}{3}<x<\frac{\mathrm{\pi }}{2}$

⇒ p(x) = cos3x is both increasing and decreasing in the interval $\left(0,\frac{\mathrm{\pi }}{2}\right)$.


Thus, the function cosx is decreasing in $\left(0,\frac{\mathrm{\pi }}{2}\right)$.

Hence, the correct answer is option (c).