Define f(x)= cases kx,&0 x<1, 2,&1 x 3

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

Problem

Define

f(x)=\begin{cases}kx,&0\le x<1,\\2,&1\le x\le3.\end{cases}

(i) Find the value of the constant

k

that makes

f

continuous on

[0,3]

. (ii) Prove that, for the value of

k

found in part (i), there exists at least one number

c

(0,3)

such that

\int_{0}^{3}f(x)\,dx=f(c)\,(3-0)

. (iii) Determine all such values of

c

explicitly.