Lagrange's theorem

Lagrange’s theorem, also called the mean value theorem, states that for a well-behaved function the average slope over the interval is realised as the instantaneous slope at at least one interior point.

Theorem — Lagrange

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists at least one $c\in(a,b)$ such that: $f'(c) = \frac{f(b)-f(a)}{b-a}.$

Remark — Geometric meaning

The right-hand side is the gradient of the secant through $(a,f(a))$ and $(b,f(b))$. The theorem says that there exists at least one point $c$ where the tangent is parallel to the secant.

At $c$ the tangent (red) is parallel to the secant (orange) joining the endpoints $(a,f(a))$ and $(b,f(b))$.

Example — Maturità exam question 2007: Lagrange on $f(x)=x^3+8$

We show that $f(x) = x^3+8$ satisfies Lagrange’s theorem on $[-2,2]$ and we find $c$.

Hypotheses: $f$ is a polynomial, hence continuous and differentiable ✓.

Computation: $\dfrac{f(2)-f(-2)}{2-(-2)} = \dfrac{(8+8)-(-8+8)}{4} = \dfrac{16}{4} = 4$.

Finding $c$: I impose $f'(c) = 3c^2 = 4 \implies c^2 = \dfrac{4}{3} \implies c = \pm\dfrac{2}{\sqrt{3}} \approx \pm 1{,}15$.

Check: $c\approx \pm 1{,}15 \in (-2,2)$ ✓.

In the simulation below you can move the endpoints $a$ and $b$: the point $c$ updates in real time and the tangent at $c$ always stays parallel to the chord $AB$.

Drag the sliders $a$ and $b$: the tangent at $c$ is always parallel to the chord $AB$.

Links

Topics: Calculus theorems
Concepts: Secant · Tangent · Mean value theorem · Lagrange’s theorem
Methods: Rolle Lagrange
People: Joseph-Louis Lagrange