Mean Value Theorem Calculator (find c)

What the theorem promises

The Mean Value Theorem is one of the bedrock results of calculus. It says that if a function f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then somewhere strictly between a and b there is a point c where the instantaneous rate of change equals the average rate of change across the whole interval:

f’(c) = (f(b) − f(a)) / (b − a)

This calculator finds that c. It computes the average rate, then solves f’(c) = average rate for every c that lands inside the interval.

The speeding-ticket intuition

Drive 150 km in 2 hours and your average speed is 75 km/h. The theorem guarantees that at some exact instant your speedometer read 75, even if you sped up and slowed down along the way. The average has to be hit at least once as an instantaneous value. Some toll roads issue tickets on exactly this logic, timing a car between two booths and inferring it must have been speeding.

The geometry

The right-hand side is the slope of the secant line joining the two endpoints of the graph. The theorem says there is at least one point where the tangent line runs parallel to that secant. Picture sliding the secant until it just touches the curve as a tangent. The touch point is c.

A worked example

Take f(x) = x² on [1, 3]. The average rate is (9 − 1) / (3 − 1) = 4. Since f’(x) = 2x, set 2c = 4, giving c = 2, which sits inside (1, 3). A cubic can produce two valid values of c in the same interval, and the calculator lists all of them.

Rolle’s theorem and the fine print

When f(a) = f(b) the average rate is zero, so the theorem guarantees a point with f’(c) = 0, a horizontal tangent. That special case is Rolle’s theorem. Both conditions matter: a sharp corner like the one in |x| at zero breaks differentiability, and the guarantee can lapse. The Mean Value Theorem quietly underlies a lot of calculus, from proving that a zero derivative everywhere means a constant function to building Taylor’s theorem.