Mean Value Theorem Calculator

This mean value theorem calculator finds every number \( c \) in \( (a, b) \) where the instantaneous rate of change \( f'(c) \) equals the average rate of change of \( f \) over \( [a, b] \). It shows the secant slope and lists all the \( c \) values it finds, in exact form when possible, so you can check the “find the c guaranteed by the MVT” problems that come up in every Calculus 1 course.

Free, instant, no sign-up.

How to use the mean value theorem calculator

1. Enter \( f(x) \). Use ^ for powers, sqrt(x), sin(x), cos(x), ln(x), e^(x) and pi. Implicit multiplication like 12x works.
2. Enter the endpoints \( a \) and \( b \) with \( a < b \). Expressions such as pi or e are accepted.
3. Press Find c. The output shows the secant slope \( \frac{f(b) - f(a)}{b - a} \) and every \( c \) strictly between \( a \) and \( b \) that satisfies the theorem (up to 8 values).

Formula

If \( f \) is continuous on \( [a, b] \) and differentiable on \( (a, b) \), the mean value theorem guarantees at least one \( c \) in \( (a, b) \) with

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Geometrically, somewhere between \( a \) and \( b \) the tangent line is parallel to the secant line through the endpoints. The calculator computes the secant slope, differentiates \( f \) exactly, and solves \( f'(x) = \text{slope} \) numerically on the open interval. Values such as \( \frac{4\sqrt3}{3} \) are recognized and shown exactly, with a decimal alongside.

Worked example

Example 1. Find the value of \( c \) for \( f(x) = \sqrt{x} \) on \( [1, 9] \).

• Secant slope: \( \frac{\sqrt9 - \sqrt1}{9 - 1} = \frac{3 - 1}{8} = \frac14 \).
• Derivative: \( f'(x) = \frac{1}{2\sqrt{x}} \).
• Solve \( \frac{1}{2\sqrt{c}} = \frac14 \): \( \sqrt{c} = 2 \), so \( c = 4 \).

Since \( 4 \) is in \( (1, 9) \), it’s the answer. That’s what the calculator above shows by default.

Example 2. Find all \( c \) for \( f(x) = x^3 - 12x \) on \( [-4, 4] \).

• \( f(4) = 64 - 48 = 16 \) and \( f(-4) = -16 \), so the secant slope is \( \frac{16 - (-16)}{8} = 4 \).
• \( f'(c) = 3c^2 - 12 = 4 \) gives \( c^2 = \frac{16}{3} \), so

$$c = \pm\frac{4}{\sqrt3} = \pm\frac{4\sqrt3}{3} \approx \pm 2.3094$$

Both values lie in \( (-4, 4) \), so there are two answers. The theorem promises at least one; the calculator lists them all. Type x^3 - 12x with \( a = -4 \), \( b = 4 \) to check.

When the calculator finds no c

The theorem only applies when its two hypotheses hold. If the calculator can’t find a \( c \), it shows a warning that continuity or differentiability may fail. For example, \( f(x) = |x| \) (type abs(x)) on \( [-1, 2] \) has secant slope \( \frac{2 - 1}{3} = \frac13 \), but \( f'(x) \) is only ever \( -1 \) or \( 1 \). The corner at \( x = 0 \) breaks differentiability, so no \( c \) exists.

Other things to keep in mind:

• Check the hypotheses yourself. A function can have a matching \( c \) even when the theorem doesn’t apply, so a result from the calculator is not proof that \( f \) is differentiable. Watch for vertical asymptotes, corners and cusps inside \( [a, b] \).
• Exam answers usually need the justification. Say that \( f \) is continuous on \( [a, b] \) and differentiable on \( (a, b) \) before quoting the \( c \) value.
• Rolle’s theorem is the special case \( f(a) = f(b) \): the secant slope is 0, so the calculator finds points with a horizontal tangent, the same points the critical points calculator looks for.

The statement, proof idea and more examples are in the mean value theorem guide. The same “average equals some actual value” idea appears for integrals in the average value of a function, and the average value calculator finds that \( c \) too.

Related guides

• Mean Value Theorem: Statement, Meaning and Worked Examples
• Derivative of √x
• Fundamental Theorem of Calculus
• Average Value of a Function

Further reading

• The Mean Value Theorem (Paul’s Online Math Notes) — Rolle’s theorem, the MVT and examples of finding \( c \).
• Continuity (Paul’s Online Math Notes) — what the continuity hypothesis means and how to check it.

FAQ

How do you find c in the mean value theorem?

Compute the secant slope \( \frac{f(b) - f(a)}{b - a} \), set \( f'(c) \) equal to it, solve for \( c \), and keep only solutions strictly between \( a \) and \( b \).

Can there be more than one value of c?

Yes. The theorem guarantees at least one, but there may be several, as in Example 2. The calculator lists every value it finds in the interval.

What if the function is not differentiable on the interval?

Then the theorem does not apply and there may be no valid \( c \). The calculator will warn you if it finds none.

What is the difference between Rolle’s theorem and the mean value theorem?

Rolle’s theorem is the case \( f(a) = f(b) \), where the guaranteed \( c \) has \( f'(c) = 0 \). The mean value theorem allows any endpoint values.