Mean Value Theorem (MVT)

If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) where:

f'(c) = [f(b) - f(a)] / (b - a)

In English: At some point, the instantaneous rate of change equals the average rate of change. The tangent line is parallel to the secant line.

AP exam usage:

"Justify that there exists a value c where f'(c) = 4." Check that f is continuous and differentiable, compute the average rate, and cite MVT.

Rolle's Theorem (special case):

If f(a) = f(b), then there exists c where f'(c) = 0. The function must have a horizontal tangent somewhere between a and b.

Free response staple. Always state the conditions (continuous on [a,b], differentiable on (a,b)) and cite MVT by name.

Reference:

Wikipedia: MVT

https://en.wikipedia.org/wiki/Mean_value_theorem