Work out mean value theorem instantly with clear inputs, formula shown and shareable results.
The mean value theorem guarantees that a function continuous on [a, b] and differentiable inside has some interior point where the instantaneous rate equals the average rate over the interval.
Mean value theorem
f'(c) = (f(b) − f(a)) / (b − a) for some c in (a, b)
The secant slope is (64 − 1)/3 = 21, and 3c² = 21 gives c = √7 ≈ 2.6457513.
If you average 60 mph over a journey, at some instant your speedometer must have read exactly 60 mph.