What is the relation between f(a) and f(b) according to Rolle’s theorem?

The correct option is "Equals to".

Rolle’s theorem is a fundamental theorem in calculus that describes the behavior of a real-valued function on a closed interval. For a function $f\left(x\right)$ defined on a closed interval $[a,b]$, the theorem states that if the following three conditions are satisfied:
1. The function $f$ is continuous on the closed interval $[a,b]$.
2. The function $f$ is differentiable on the open interval $(a,b)$.
3. The values of the function at the endpoints of the interval are equal, meaning:
$$f\left(a\right)=f\left(b\right)$$
Then, there exists at least one number $c$ in the open interval $(a,b)$ such that the derivative at that point is zero:
$$f^{\prime }\left(c\right)=0$$

Geometrically, this means that if a smooth curve starts and ends at the exact same height (y-value) over an interval, there must be at least one point along the curve where the tangent line is horizontal (the slope is zero).

Therefore, the essential relation required between $f\left(a\right)$ and $f\left(b\right)$ to satisfy Rolle's theorem is that they must be equal to each other.