Lagrange mean value theorem

Statement

Suppose $f$ is a function defined on a closed interval $[a,b]$ (with $a < b$) such that the following two conditions hold:

1. $f$ is a continuous function on the closed interval $[a,b]$ (i.e., it is right continuous at $a$, left continuous at $b$, and two-sided continuous at all points in the open interval $(a,b)$).
2. $f$ is a differentiable function on the open interval $(a,b)$, i.e., the derivative exists at all points in $(a,b)$. Note that we do not require the derivative of $f$ to be a continuous function.

Then, there exists $c$ in the open interval $(a,b)$ such that the derivative of $f$ at $c$ equals the difference quotient $\Delta f(a,b)$. More explicitly:

$f'(c) = \frac{f(b) - f(a)}{b - a}$

Geometrically, this is equivalent to stating that the tangent line to the graph of $f$ at $c$ is parallel to the chord joining the points $(a,f(a))$ and $(b,f(b))$.

Note that the theorem simply guarantees the existence of $c$, and does not give a formula for finding such a $c$ (which may or may not be unique).

Related facts

• Rolle's theorem
• Zero derivative implies locally constant
• Fundamental theorem of calculus
• Positive derivative implies increasing
• Increasing and differentiable implies nonnegative derivative