GATE CS 1987 Q1-xxvi — Maxima and Minima, Mean Value Theorem, Integration

Under the standard assumption that $f(x)$ is a continuous function on the interval $[x_i, x_{i+1}]$, we can apply the Intermediate Value Theorem (IVT).

The Intermediate Value Theorem states that if a function $f(x)$ is continuous on a closed interval $[a, b]$, and $u$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c \in (a, b)$ such that $f(c) = u$.

Given that $f(x_i) \cdot f(x_{i+1}) < 0$, the function values at the endpoints have opposite signs (one is positive and the other is negative). Since $0$ lies strictly between these two values, there must exist at least one root $c \in (x_i, x_{i+1})$ such that:

Thus, there must be a root of $f(x)$ between $x_i$ and $x_{i+1}$. This principle forms the basis of root-finding algorithms like the Bisection Method.

Options C and D are incorrect because the signs of the function values at the endpoints do not provide any information about the vanishing of the fourth derivative.