Let be a piecewise–defined function that is continuous on the closed interval

MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let

f

be a piecewise–defined function that is continuous on the closed interval

[2,8]

. Which of the following statements must be true about the average value of

f

[2,8]

? (a) The average value of

f

[2,8]

\displaystyle\int_{2}^{8}f(x)\,dx

. (b) The average value of

f

[2,8]

must lie strictly between

f(2)

and

f(8)

. (c) There exists at least one number

c

(2,8)

such that

f(c)

equals the average value of

f

[2,8]

. (d) The average value of

f

[2,8]

\dfrac{f(2)+f(8)}{2}