Two functions are given on the interval : f_1(x)=|x|

MATHEMATICS · Subjective · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Two functions are given on the interval

[-1,1]

f_1(x)=|x|,\qquad f_2(x)=\begin{cases}1,&x\neq0\\0,&x=0.\end{cases}

(i) Identify which of the two functions satisfies every hypothesis required for the Mean Value Theorem for Integrals on

[-1,1]

. Give a reason for your answer. (ii) For each function separately, explain whether a number

c\in(-1,1)

can be guaranteed such that

\int_{-1}^{1} f(x)\,dx=f(c)\,(1-(-1))

. Justify your reasoning without evaluating any integrals.