For a function that is continuous on and differentiable on

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

For a function that is continuous on

[a,b]

and differentiable on

(a,b)

, which statement is always true? (a)

\displaystyle f'(x)=\frac{f(b)-f(a)}{b-a}\text{ for every }x\in(a,b)

(b)

\displaystyle f'(a)=\frac{f(b)-f(a)}{b-a}

(c)

\displaystyle \text{There exists }c\in(a,b)\text{ such that }f'(c)=\frac{f(b)-f(a)}{b-a}

(d)

\displaystyle \text{There exists }c\in[a,b]\text{ such that }f'(c)=\frac{f(b)-f(a)}{b-a}