Animated Solution for Mathematics - Differentiation: For all twice differentiable functions $f:\mathbb{R}\to \mathbb{R}$, with $f(0)=f(1)=f^{\prime }(0)=0$

Imagine you are standing on a graph, tracing the path of a particle. You are given a function $f:\mathbb{R}\to \mathbb{R}$ that is twice differentiable. This implies the path is incredibly smooth—no sharp turns, no sudden jumps, just a graceful, flowing curve.

We are given three anchors: $f(0)=0$, $f(1)=0$, and $f^{\prime }(0)=0$. These values define the behavior of our curve. The first two indicate the particle starts at the origin and returns to the $x$-axis at $x=1$. The third, $f^{\prime }(0)=0$, confirms that at the start, the particle is moving perfectly horizontally.

Finding the Peak

We want to investigate the second derivative, $f^{\prime \prime }(x)$. Before we tackle that, let us examine the first derivative. We know $f(0)=0$ and $f(1)=0$.

This is the classic setup for Rolle's Theorem. If a continuous and differentiable function starts and ends at the same height, it must have a point in between where the slope is zero.

Therefore, there exists some $c_1\in (0,1)$ such that:

Geometrically, this is the moment the particle reaches a peak or a valley. It must stop climbing or diving to turn around and head back to the $x$-axis. We have now identified a second point where the slope is zero.

The Heart of the Matter

Now, let us treat the first derivative, $f^{\prime }(x)$, as a function in its own right. We know two things about this new function: first, $f^{\prime }(0)=0$ (given), and second, $f^{\prime }(c_1)=0$ (which we just discovered).

Consider the interval $[0,c_1]$. Our function $f^{\prime }(x)$ is continuous and differentiable because $f(x)$ is twice differentiable. Since it takes the same value, zero, at both $x=0$ and $x=c_1$, Rolle's Theorem applies once more.

By applying Rolle's Theorem to $f^{\prime }(x)$ on the interval $[0,c_1]$, we are guaranteed the existence of a point $c_2\in (0,c_1)$ such that the derivative of $f^{\prime }(x)$ is zero. Since the derivative of $f^{\prime }(x)$ is $f^{\prime \prime }(x)$, we conclude:

A Mathematical Victory

We have successfully navigated the terrain. We found that $f^{\prime \prime }(c_2)=0$ for some $c_2$ in the interval $(0,c_1)$. Since $c_1<1$, our point $c_2$ is safely tucked away inside the open interval $(0,1)$.

We have proven that there must be at least one point where the second derivative vanishes. This is a fundamental truth about the nature of smooth motion.

Whenever you see a function that starts flat and returns to its starting height, you now know that somewhere in that journey, the curvature must vanish. Keep this logic in your toolkit—it is one of the most powerful ways to visualize the hidden behavior of functions in JEE Advanced.