Animated Solution for Mathematics - Differentiation: Let $f(x)=2+\cos x$ for all real $x$. \\ STATEMENT-1: For each real $t$, there exists a point $c$ in $[t,t+\pi ]$ such that $f^{\prime }(c)=0$ because \\ STATEMENT-2: $f(t)=f(t+2\pi )$ for each real $t$.

Imagine you are standing on a calm, rolling ocean. The surface of the water follows a rhythmic pattern described by the function $f(x)=2+\cos x$.

This is a cosine wave shifted upwards by two units. Because it is a simple cosine wave, it is perfectly smooth, continuous, and differentiable everywhere. There are no sharp corners or sudden breaks—just a gentle, infinite oscillation.

Finding the Peaks and Valleys

To understand the behavior of this function, we look at its rate of change. When we differentiate $f(x)=2+\cos x$ with respect to $x$, the constant $2$ vanishes, and the derivative of $\cos x$ becomes $-\sin x$.

We obtain the derivative function:

This function $f^{\prime }(x)$ represents the slope of the tangent to our original curve at any point $x$. When $f^{\prime }(x)=0$, the tangent is perfectly horizontal, identifying the peaks and valleys of our wave.

Statement-1 considers any interval $[t,t+\pi ]$. Think of this as a sliding window of fixed length $\pi$ moving along the $x$-axis. We want to determine if there is always a point $c$ in this window where $f^{\prime }(c)=0$.

The roots of our derivative are found by solving:

From trigonometry, we know $\sin c=0$ at every integer multiple of $\pi$, specifically $c=n\pi$ for $n\in \mathbb{Z}$. These roots occur at $0,\pi ,2\pi ,3\pi ,\dots$.

The distance between any two consecutive roots is exactly $\pi$. If you place a window of length $\pi$ anywhere on the $x$-axis, it is mathematically impossible to avoid these roots; you are guaranteed to capture at least one of them. Thus, Statement-1 is true.

Now, consider Statement-2: $f(t)=f(t+2\pi )$. This is a fundamental property of the cosine function. Since $\cos (t+2\pi )=\cos t$, it follows that:

This confirms that the function is periodic with a period of $2\pi$. Therefore, Statement-2 is also true.

However, we must determine if Statement-2 is the correct explanation for Statement-1. Does a period of $2\pi$ automatically guarantee a root of the derivative in every interval of length $\pi$?

Consider a function with a period of $2\pi$ that remains very flat for most of the interval and changes rapidly only in a small region. Such a function would satisfy the periodicity condition but might not have a derivative root in every sub-interval of length $\pi$.

Rolle's Theorem only guarantees a root in an interval of length $2\pi$ (the period). It does not provide the guarantee for a smaller interval of length $\pi$.

The truth of Statement-1 relies on the specific, rigid spacing of the roots of the sine function, not just the general property of periodicity. Therefore, while both statements are true, Statement-2 is not the correct explanation for Statement-1.