Mean Value Theorem calculator

The Mean Value Theorem (MVT) states that for a function continuous on a closed interval [a, b] and differentiable on the open interval (a, b), there is at least one point c where the instantaneous rate of change f ‘(c) equals the average rate of change across the whole interval. This calculator works out that average rate and finds the value(s) of c for you.

The formula

The theorem says there exists c in (a, b) such that f ‘(c) = [f(b) − f(a)] ÷ (b − a). Geometrically, this means somewhere on the curve the tangent line is parallel to the straight line (secant) joining the endpoints. The calculator computes the right-hand side, then searches the interval for every c where the derivative matches it.

When the theorem applies

The MVT requires the function to be continuous on [a, b] and differentiable on (a, b). If your function has a break or sharp corner in the interval, a valid c may not exist. To see the curve and its secant line, plot it with our graph plotter, and use the calculus solver for derivatives and integrals.

Frequently asked questions

Can there be more than one value of c?

Yes. The theorem guarantees at least one, but some functions have several. This calculator reports all the values it finds in the interval.