What is the formula for Lagrange’s theorem?

The correct option is f’(c) = f(b)−f(a)/b−a.

Lagrange's Mean Value Theorem (often simply called the Mean Value Theorem) is a fundamental theorem in calculus that relates the average rate of change of a function over an interval to the instantaneous rate of change of the function at some point within that interval.

Specifically, if a function $f\left(x\right)$ satisfies the following two conditions:
1. It is continuous on the closed interval $[a,b]$.
2. It is differentiable on the open interval $(a,b)$.

Then, there exists at least one point $c$ in the open interval $(a,b)$ such that the derivative of the function at $c$, denoted as $f^{\prime }\left(c\right)$, is equal to the average rate of change of the function over $[a,b]$.

Mathematically, the average rate of change (which represents the slope of the secant line connecting the points $(a,f(a\left)\right)$ and $(b,f(b\left)\right)$) is given by the formula:

$\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Lagrange's theorem states that at some point $c$, the instantaneous rate of change (the slope of the tangent line) equals this average rate of change:
$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Therefore, the formula is expressed as f’(c) = f(b)−f(a)/b−a.