The Rolle’s theorem states that the derivative of any real-valued differentiable function attaining equal values at two distinct points will vanish at some point between them. More precisely, let \(f\) be a real-valued function continuous on \([a,b]\) and differentiable on \((a,b)\) with \(f(a) = f(b)\). Then, the Rolle’s theorem states that there is a point \(c \in (a,b)\) such that \(f'(c) = 0\).

#### A proof of the Rolle’s theorem

**Result** (Rolle’s theorem). Let \(f\) be a real-valued function continuous on \([a,b]\) and differentiable on \((a,b)\) with \(f(a) = f(b)\). Then, there is a point \(c \in (a,b)\) such that \(f'(c) = 0\).

The proof is as follows:

Suppose that \(f(c) \neq 0\) for all \(c \in (a,b)\). Since \(f\) is continuous on \([a,b]\), by the extreme value theorem, \(f\) attains its extrema on \([a,b]\). Since \(f\) is differentiable on \((a,b)\), by the first derivative theorem, \(f\) cannot attain its extrema at any point in \((a,b)\). Therefore, \(f\) attains its extrema at the points \(a\) and \(b\). Now, since \(f(a) = f(b)\), it follows that \(f\) is a constant function and its derivative is zero everywhere which is obviously a contradiction. This completes the proof.

#### Applications of the Rolle’s theorem

By definition, a polynomial \(p(x) = \sum_{i=0}^{n} a_i x^i\) splits if $$p(x) = k \prod_{i=1}^{n} (x-r_i),$$ where \(k\) and \(c_i\)s are real numbers. We use the Rolle’s theorem to prove the following result:

**Result**. If a real polynomial \(p\) of degree \(n > 1\) splits, then so does its derivative \(p’\).

The proof is as follows:

Let \(p\) split. Then \(p\) has exactly \(n\) roots. Now, suppose that the \(n\) roots \({r_i}_{i=1}^{n}\) of \(p\) are distinct. Without loss of generality, we may assume that $$r_1 < r_2 < \dots < r_n.$$ By the Bolzano’s theorem (proved in the post on the intermediate value theorem), the polynomial \(p’\) which is of degree \(n -1\) has \(n-1\) roots \(c_i\)s such that $$a_i < c_i < a_{i+1}$$ for each \(1 \leq i \leq n-1\). It is up to the reader to prove that if \(p\) has \(n\) roots and some of them are repeated, then, again, \(p’\) has \(n-1\) roots and so, \(p’\) splits.

**Exercise**. Show that the polynomial function $$f(x) = x^5 – 4x + 2$$ has exactly three real roots.

**Solution**. Observe that \(f(-2) < 0\), \(f(0) > 0\), \(f(1) = -1\), and \(f(2) > 0\). This shows that \(f\) has at least three real roots by Rolle’s theorem. On the other hand, since $$f'(x) = 5x^4 – 4$$ has exactly two real roots, by the above result, \(f\) cannot have more than three roots. Therefore, \(f\) has exactly three real roots.