Rolle’s Theorem
If f(a)=f(b) there is a point where f′=0.
Key idea
If f(a)=f(b) there is a point where f′=0.
In depth
If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f′(c) = 0 for some c in (a, b). It is the special case of the MVT with a flat secant, guaranteeing a horizontal tangent.
Worked example
f(x) = x² − 4x on [0, 4]: f(0) = f(4) = 0, f′(2) = 0.
Real-world uses
- Justifying why a ball thrown up and down must stop rising at its highest point
Common mistake
Forgetting the requirement f(a) = f(b).
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Rolle’s Theorem?
If f(a)=f(b) there is a point where f′=0. It is a topic at the expert level in Calculus.
Where is Rolle’s Theorem used in real life?
Real-world examples: Justifying why a ball thrown up and down must stop rising at its highest point.
What should I learn before Rolle’s Theorem?
Make sure you are comfortable with Mean Value Theorem, the previous topic in Calculus.
What is a common mistake with Rolle’s Theorem?
Forgetting the requirement f(a) = f(b).
How can I practise Rolle’s Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Rolle’s Theorem?
Move on to L’Hôpital’s Rule next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.