# Which c fits the mean value theorem?

Finds every c in (a, b) where f′(c) = (f(b) − f(a))/(b − a).

- Page: https://www.acalculator.org/math/mean-value-theorem-calculator
- JSON spec: https://www.acalculator.org/math/mean-value-theorem-calculator.json
- Version: 5e7dd71f25d5

## Default answer

Example with the default inputs (Function f(x) sqrt(x), From a 0, To b 9): For sqrt(x) on [0, 9], c = 9/4.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(x) | The function of x. |
| a | From a | Left end. |
| b | To b | Right end. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| c | Values of c | Each c in (a, b) where f′(c) is the slope. |
| slope | Slope of the secant | (f(b) − f(a))/(b − a). |

## Method

f′(c) = (f(b) − f(a))/(b − a) for some c in (a, b).

## Assumptions

- Angles in radians.

## Worked examples

1. f = sqrt(x), a = 0, b = 9 gives c = 9/4, slope = 1/3. Source: OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.4 The Mean Value Theorem, Example 4.15.
2. f = x^3 - 4x, a = -2, b = 2 gives c = -2 sqrt(3)/3, 2 sqrt(3)/3, slope = 0. Source: OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.4 The Mean Value Theorem, Example 4.14(b).

## FAQ

### What does the mean value theorem say?

If f is continuous on [a, b] and differentiable on (a, b), there is at least one c in (a, b) with f′(c) = (f(b) − f(a))/(b − a). Somewhere between a and b, the instantaneous rate of change equals the average rate of change.

### What is Rolle's theorem?

The special case f(a) = f(b): then the secant is horizontal and some c in (a, b) has f′(c) = 0. For x³ − 4x on [−2, 2], f(−2) = f(2) = 0, and f′(c) = 3c² − 4 = 0 at c = ±2/√3.

### How do I find the value of c?

Work out the secant slope m = (f(b) − f(a))/(b − a), then solve f′(c) = m for c and keep the solutions strictly between a and b. For √x on [0, 9]: m = (3 − 0)/9 = 1/3, and 1/(2√c) = 1/3 gives c = 9/4.

### What if the theorem does not apply?

If f has a jump or a pole in [a, b], or a corner in (a, b), there may be no such c. For |x| on [−1, 1] the secant slope is 0 but f′ is ±1 everywhere except 0, where it does not exist. The page names what fails instead of looking for c.

### Can there be more than one c?

Yes: the theorem promises at least one. For sin(x) on [0, 2π] the secant slope is 0 and cos(c) = 0 at both π/2 and 3π/2. The page lists them all.

### How is the answer checked?

The derivative from the computer algebra system is compared with a numeric difference quotient at 20 points. Each c is found both by the algebra and by a sign scan on a fine grid of (a, b), and each c the algebra gives is checked by putting it back into f′(x) − m.

## Sources

- OpenStax, Calculus Volume 1, section 4.4 The Mean Value Theorem: https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem