MATH1131 Chap.9 Mean Value Theorem and L'Hôpital's Rule
Mean Value Theorem and L'Hôpital's Rule
The mean value theorem says that a function with a derivative at every interior point of [a, b], and continuous on the whole closed interval, has, somewhere strictly inside the interval, a tangent parallel to the chord joining the endpoints. Its special case with equal endpoint values is Rolle's theorem.
Although it only asserts that a suitable point c exists, the theorem has wide consequences that the course uses repeatedly: a zero derivative on an interval forces a constant function, a positive derivative forces a strictly increasing function, and two antiderivatives of one function on an interval differ by a constant.
It also proves inequalities and error bounds by controlling the derivative over the interval rather than finding c. The chapter covers the first and second derivative tests for classifying stationary points, finding global extremes on closed intervals by comparing stationary points, non-differentiable points and endpoints, and counting zeros through Rolle's theorem.
It finishes with l'Hôpital's rule for limits of the forms zero over zero and infinity over infinity, together with the rewriting needed for other indeterminate forms.
What this chapter covers
- 01
Statement and hypotheses of the mean value theorem
- 02
Rolle's theorem and counting zeros
- 03
Increasing, decreasing and constant functions
- 04
Proving inequalities and error bounds
- 05
First and second derivative tests
- 06
Global extremes on closed intervals
- 07
Antiderivatives differ by a constant
- 08
L'Hôpital's rule and other indeterminate forms
Worked example · free
Global extremes on a closed interval
- 1f is a polynomial, hence continuous on the closed interval, so both extremes exist by the maximum-minimum theorem.
- 1f'(x) = 3x^2 - 3 = 0 at x = -1 and x = 1, both inside the interval.
- 1Evaluate the candidates: f(-2) = -1, f(-1) = 3, f(1) = -1, f(3) = 19.
- 1Compare: the global maximum is 19 at x = 3 and the global minimum is -1, attained at both x = -2 and x = 1.
Key terms
- Mean value theorem
- The result that a function continuous on [a, b] and differentiable inside has a point where the derivative equals the chord slope.
- Rolle's theorem
- The special case of the mean value theorem in which the endpoint values are equal, giving a zero derivative inside.
- Critical point
- An interior point where the derivative is zero or does not exist.
- Global maximum
- The largest value a function attains on a given set.
- Antiderivative
- A function whose derivative is the given function on an interval.
- Indeterminate form
- A limit pattern such as zero over zero whose value cannot be read from the limits of its parts.
Mean Value Theorem and L'Hôpital's Rule FAQ
Do I need to find c when using the mean value theorem in a proof?
Usually not. Proofs use only the fact that c lies strictly inside the interval, which bounds the derivative at c. Computing c is required only when a question explicitly asks for it.
Which candidates must I check for a global maximum on a closed interval?
Check stationary points, points where the derivative does not exist, and both endpoints, then compare the function values. The largest is the global maximum and the smallest the global minimum.
When is l'Hôpital's rule allowed?
When the limit has the form zero over zero or infinity over infinity and both functions are differentiable near the point. Differentiate numerator and denominator separately, not with the quotient rule, and check the form again before each further use.
How do I handle a limit of the form zero times infinity?
Rewrite the product as a quotient by moving one factor into the denominator as its reciprocal. That produces a zero over zero or infinity over infinity form to which l'Hôpital's rule applies.
What does the second derivative test do when f'' is zero?
It gives no conclusion, since such a point can be a maximum, a minimum or neither. Use the first derivative test instead, examining the sign of f' on each side of the stationary point.
How does Rolle's theorem help count the zeros of a function?
Between any two zeros of a differentiable function there is a zero of its derivative. So if the derivative has at most k zeros, the function has at most k plus one zeros, which bounds the number of roots.
Exam move
Learn the exact statement of the mean value theorem, including both hypotheses, and practise writing it in one sentence because exams ask for it. Then work through inequality and estimate proofs until the pattern is familiar: choose the function and interval, apply the theorem, bound the derivative using the range of c, and conclude. For extremes, use a small table of candidates and values.
For l'Hôpital questions, first try factorising or dividing by a dominant term, and record the indeterminate form at each application. When a proof question asks you to show an inequality for all positive x, look for a function equal to zero at x equal to zero whose derivative has a fixed sign.
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