Lagrange’s mean value theorem is also called as

The correct option is "the mean value theorem".

Step-by-Step Explanation:

1. Definition of Lagrange's Mean Value Theorem:
Lagrange's Mean Value Theorem (often abbreviated as LMVT) states that if a function $f$ is continuous on a closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists at least one point $c$ in the open interval $(a,b)$ such that:

$$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

2. Terminology:
In calculus, when mathematicians refer to "the mean value theorem" without any prefix, they are referring to Lagrange's formulation. Rolle's theorem is actually a special case of Lagrange's mean value theorem where $f\left(a\right)=f\left(b\right)$, resulting in $f^{\prime }\left(c\right)=0$. Therefore, Lagrange's mean value theorem is also widely known simply as the mean value theorem.