Animated Solution for Mathematics - Differentiation: Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=\{\begin{array}{ll}(2-\sin (\frac{1}{x}))|x|,& x\ne 0\\ 0,& x=0\end{array}$. Then $f$ is

Welcome, fellow traveler on the path to JEE mastery. Today, we are going to dissect a function that, at first glance, looks like a simple geometric construction but hides a chaotic, beautiful truth within its core.

We are looking at $f(x)=(2-\sin (\frac{1}{x}))|x|$. It is a function that dares to challenge our intuition about what it means to 'increase' or 'decrease.'

When we see an absolute value like $|x|$, our first instinct should be to strip it away. We are dealing with a piecewise reality.

For $x>0$, our function simplifies to $f(x)=x(2-\sin (\frac{1}{x}))$. For $x<0$, it becomes $f(x)=-x(2-\sin (\frac{1}{x}))$.

Imagine the graph. The term $(2-\sin (\frac{1}{x}))$ is a rhythmic, bounded pulse. Since the sine function is trapped between $-1$ and $1$, the term $(2-\sin (\frac{1}{x}))$ is always trapped between $1$ and $3$.

It never touches zero, and it never explodes. It is a steady, oscillating heartbeat. When we multiply this by $x$, we are essentially stretching this heartbeat between the lines $y=x$ and $y=3x$. It looks like a wave being squeezed into a funnel as it approaches the origin.

To determine if a function is monotonic, we must ask: does it ever turn back? Does it ever change its mind about which way it is going? This is the domain of the derivative, $f^{\prime }(x)$.

Let us focus on the positive side, $x>0$. We apply the Product Rule to $f(x)=x\cdot (2-\sin (\frac{1}{x}))$. Using the rule $\frac{d}{dx}(uv)=u^{\prime }v+u{v}^{\prime }$, we get:

Now, let us tackle that second term with the Chain Rule. The derivative of $2$ is $0$, and the derivative of $-\sin (\frac{1}{x})$ involves the derivative of the inner function $\frac{1}{x}$, which is $-\frac{1}{x^2}$. Thus:

Putting it all together, our derivative becomes:

Here is where the magic—and the trap—lies. Look closely at the expression for $f^{\prime }(x)$. We have two distinct characters in this play.

The first part, $2-\sin (\frac{1}{x})$, is a well-behaved, bounded actor. It stays between $1$ and $3$. But the second part, $\frac{1}{x}\cos (\frac{1}{x})$, is a wild, untamed force.

As $x$ approaches $0$ from the right, $\frac{1}{x}$ explodes toward infinity. The $\cos (\frac{1}{x})$ term continues to oscillate between $-1$ and $1$.

When you multiply an infinite growth by an oscillation, you get an oscillation that grows to infinite amplitude. Near the origin, this term completely dominates the expression. It will force $f^{\prime }(x)$ to swing from massive positive values to massive negative values, crossing zero infinitely many times in any interval $(0,ϵ)$.

Because the derivative $f^{\prime }(x)$ changes sign infinitely often, the function cannot be monotonic. It is constantly 'wiggling' up and down, never committing to a single direction. By symmetry, the same logic applies to the negative side of the domain.

I know this might feel counter-intuitive. We often want functions to be 'nice' and 'smooth.' But in the world of JEE Advanced, we must embrace the 'wiggly' functions.

They teach us that even when a function is continuous, its behavior can be incredibly complex. You have successfully navigated the trap of assuming monotonicity based on the envelope, and you have used the power of the derivative to reveal the truth. Keep this rigor in your toolkit—it is exactly what separates the good from the great.