What are/is the conditions to satify Lagrange’s mean value theorem?

The correct option is "f is differentiable and continuous on (a,b)".

Lagrange's Mean Value Theorem (LMVT) states that if a real-valued function $f$ is defined on a closed interval $\left[a,b\right]$, it must satisfy the following fundamental conditions:
1. The function $f$ must be continuous on the closed interval $\left[a,b\right]$.
2. The function $f$ must be differentiable on the open interval $\left(a,b\right)$.

To understand why the correct option specifies that the function must be both differentiable and continuous on the open interval $\left(a,b\right)$, we look at the relationship between differentiability and continuity:
A fundamental theorem in calculus states that if a function is differentiable at any point, it must also be continuous at that point. Therefore, because the function $f$ is required to be differentiable on the open interval $\left(a,b\right)$, it automatically implies that $f$ is also continuous on the same open interval $\left(a,b\right)$.

Thus, on the open interval $\left(a,b\right)$, the function is both differentiable and continuous, making this option the correct statement describing the behavior of the function within the open interval.