Lecture: Rolle’s Theorem

What is Rolle’s Theorem?

Rolle’s Theorem is a special case of the Mean Value Theorem (MVT).
It guarantees that if a continuous and differentiable function starts and ends at the same height, there must be at least one point between them where the slope of the tangent line is zero (that is, $$f^\prime(c) = 0$$.

The Formal Statement

Let $$f(x)$$ be a function such that:

1. $$f(x)$$ is continuous on the closed interval [a,b],

2. $$f(x)$$ is differentiable on the open interval (a,b),

3. $$f(a) = f(b)$$.

Then there exists at least one number c∈(a,b) such that:

$$f^\prime(c) = 0$$

Intuitive Meaning

Rolle’s Theorem says:
If you draw a smooth curve that begins and ends at the same height, the curve must have at least one horizontal tangent line somewhere in between.

That means the function must reach a peak or a valley (or flatten out) between a and b.

Visual Explanation

Imagine the curve passing through A(a,f(a)) and B(b,f(b)) at the same height.

Then, according to Rolle’s Theorem:

• The tangent at some point C(c,f(c)) between A and B is horizontal.

• So, $$f^\prime(c) = 0$$.

Example 1

Let $$f(x) = x^2-4x+3$$ on the interval [1,3].

Step 1. Check conditions:

• Continuous on [1,3]: ✅ (polynomial)

• Differentiable on (1,3): ✅

• $$f(1)=0 , f(3)=0 \longrightarrow f(1)=f(3)$$: ✅

Step 2. Compute $$f^\prime(x)$$ :

$$f^\prime(x) = 2x – 4$$

Step 3. Find $$f^\prime(c) = 0$$:

&&f^\prime(c) = 2c – 4 = 0 \longrightarrow c = 2&&

✅ Therefore, c=2 satisfies Rolle’s Theorem.

Example 2

Let $$f(x)=\sin(x)$$ on $$[0 , \pi]$$.

Step 1. Check conditions:

• Continuous and differentiable: ✅

• $$f(0) = 0 , f(\pi) = 0 \longrightarrow f(0) = f(\pi)$$: ✅

Step 2. Compute $$f^\prime(x)$$:

$$f^\prime(x) = \cos(x)$$

Step 3. Find $$f^\prime(c) = 0$$:

$$f^\prime(c) = \cos(c) = 0\longrightarrow c = \frac{\pi}{2}$$​

✅ So $$f^\prime(c)=0 at c=\frac{\pi}{2}$$

💡 8. Geometric Interpretation