Answer: f(b)-f(a) = f'(c)(b-a) for some c in (a,b).
- A f(c) = 0 for some c
- B f(b)-f(a) = f(c)(b-a) for some c in (a,b)
- C f(b)-f(a) = f'(c)(b-a) for some c in (a,b)
- D f is monotone
Correct answer: C. f(b)-f(a) = f'(c)(b-a) for some c in (a,b)
Explanation: MVT (Lagrange): if f is continuous on [a,b] and differentiable on (a,b), there exists c in (a,b) such that f'(c) = [f(b)-f(a)]/(b-a).
As point Q slides along the curve toward P, the secant line PQ rotates into the tangent line at P, whose slope is the derivative.
Concept context
Rate of change, differentiation rules, and applications