GATE CS 1995 Q1.21 — Maxima and Minima, Mean Value Theorem, Integration

To find the number of solutions to the equation $x = \cos x$ in the interval $[0, \pi]$, let us define the function:

$$f(x) = x - \cos x$$

First, we evaluate the function at the boundaries of the interval $[0, \pi]$:

• At $x = 0$: $f(0) = 0 - \cos(0) = -1 < 0$
• At $x = \pi$: $f(\pi) = \pi - \cos(\pi) = \pi + 1 > 0$

Since $f(x)$ is continuous on $[0, \pi]$ and changes sign, by the Intermediate Value Theorem, there exists at least one root in the interval $(0, \pi)$.

To check uniqueness, we find the derivative of $f(x)$:

$$f'(x) = 1 + \sin x$$

For any $x \in [0, \pi]$, $\sin x \ge 0$, which means $f'(x) \ge 1 > 0$. Since $f'(x) > 0$ throughout the interval, $f(x)$ is strictly increasing. A strictly increasing function can cross the x-axis at most once. Thus, there is exactly one solution.