Lagrange's Theorem

Statement

Verify Lagrange’s theorem (mean value) for $f(x) = \sqrt{x}$ on $[0,4]$ and find $c$.

Solution

Hypotheses: $f(x)=\sqrt{x}$ is continuous on $[0,4]$ and differentiable on $(0,4)$ (at $0$ the derivative is infinite, but the open interval excludes it). Lagrange’s hypotheses are satisfied.

Thesis: there exists $c\in(0,4)$ with $f'(c)=\dfrac{f(4)-f(0)}{4-0}=\dfrac{2-0}{4}=\dfrac12$.

From $f'(x)=\dfrac{1}{2\sqrt{x}}=\dfrac12$ it follows that $\sqrt{c}=1$, that is $\boxed{c=1}\in(0,4)$.

Links

Topics: Derivatives
Concepts: Derivative · Lagrange’s theorem
Skills: Proving
People: Joseph-Louis Lagrange
Exercise type: Proof