Question

If f(x) = |x| + |x - 1| + |x - 2|, then

• A. f (x) has minima at x = 1
• B. f (x) has maxima at x = 0
• C. f (x) has maxima or minima at x = 3
• D. None of these

Answer 1

The correct option is A

f (x) has minima at x = 1

We have,
f(x) = |x| + |x - 1| + |x - 2|, = $\{\begin{array}{cc}-3x+3,& x<0\\ -x+3,& 0\le xv1\\ x+1,& 1\le x<2\\ 3x-3,& x\ge 2\end{array}\Rightarrow f^{\prime }\left(x\right)=\{\begin{array}{cc}-3& x<0\\ doesnotexist& x=0\\ -1& 0<x<1\\ doesnotexist& x=1\\ 1& 1<x<2\\ doesnotexist& x=2\\ 3& x>2\end{array}$
Clearly, f(x) has minima at x = 1 and neither maxima r. Minima at x = 0, x = 1 and x = 2
Hence (a) is the correct answer.