Animated Solution for Mathematics - Differentiation: Let $f(x)=x|x|$. The set of points where $f(x)$ is twice differentiable is .........

Imagine you are standing on a coordinate plane, looking at the function $f(x)=x|x|$. At first glance, it looks like a simple, harmless curve.

In the world of JEE Advanced, appearances can be deceiving. This function is a classic trap, a beautiful example of how a function can be smooth enough to be differentiated once, yet fail the test of second-order differentiability.

To truly understand $f(x)=x|x|$, we must strip away the absolute value. The absolute value function $|x|$ is a chameleon; it changes its identity based on the sign of $x$.

For $x\ge 0$, $|x|=x$, so our function becomes:

For $x<0$, $|x|=-x$, so our function becomes:

We have effectively stitched two parabolas together at the origin. One opens upward for positive $x$, and one opens downward for negative $x$.

Now, let us find the first derivative, $f^{\prime }(x)$. For $x>0$, the derivative of $x^2$ is $2x$. For $x<0$, the derivative of $-x^2$ is $-2x$.

If we combine these, we get $f^{\prime }(x)=2|x|$. But what happens at the origin? We cannot just assume it is differentiable; we must use the limit definition:

Calculating the left-hand derivative (LHD) and right-hand derivative (RHD), we find that both approach $0$. Since $LHD=RHD=0$, the function is indeed differentiable at $x=0$, and $f^{\prime }(0)=0$.

The graph of $f^{\prime }(x)=2|x|$ is a sharp 'V' shape, perfectly smooth at the origin.

Now, we reach the climax of our story. We want to find the second derivative, $f^{\prime \prime }(x)$. We differentiate $f^{\prime }(x)=2|x|$.

For $x>0$, the derivative of $2x$ is $2$. For $x<0$, the derivative of $-2x$ is $-2$. So, $f^{\prime \prime }(x)$ is $2$ for $x>0$ and $-2$ for $x<0$.

But what about $x=0$? Let us check the limit of $f^{\prime \prime }(x)$ as we approach zero:

Because the left-hand limit and right-hand limit of the second derivative do not match, $f^{\prime \prime }(0)$ does not exist. The first derivative $f^{\prime }(x)=2|x|$ has a 'kink' or a sharp corner at the origin.

Just as a sharp corner in a function prevents the first derivative from existing, a sharp corner in the first derivative prevents the second derivative from existing.

We have discovered that $f(x)=x|x|$ is differentiable everywhere, but it is only twice differentiable on the set $\mathbb{R}-\{0\}$. The origin is the point of failure.

This problem teaches us that differentiability is a layered concept. A function can be smooth, but its derivative might not be. Keep this in mind as you tackle more complex calculus problems; always respect the limit definition, especially at the points where the function's behavior changes.