Question

Prove that f(x) = x+sinx is an increasing function
for all x real​

Answer 1

Given,

$\longrightarrow f(x)=x+\sin x$

We need to prove that $f(x)$ is strictly increasing $\forall x\in\mathbb{R}.$

Taking derivative of $f(x)$ wrt $x,$

$\longrightarrow f'(x)=1+\cos x$

We know that,

• $1+\cos x=2\cos^2\left(\dfrac{x}{2}\right)$

Thus,

$\longrightarrow f'(x)=2\cos^2\left(\dfrac{x}{2}\right)$

Since $x^2\geq 0\quad\!\forall x\in\mathbb{R},$

$\longrightarrow f'(x)\geq 0$

So $f'(x)$ is always non - negative $\forall x\in\mathbb{R}.$ This implies $f(x)$ is increasing $\forall x\in\mathbb{R}.$

Thus $f(x)$ is an increasing function.

Hence Proved!