Question

A function satisfying the conditions of Lagrange's mean value theorem on the interval $[1, 5]$ is:

• A. $f(x) = |x - 1| + |x - 5|$
• B. $g(x) = x - [x]$ ($[x]$ represents integral part of $x$)
• C. $h(x) = \begin{cases} 3x - 1, & 0 \le x \le 1 \\ x^2 + 1, & 1 \le x \le 4 \\ 17, & 4 \le x \le 6 \end{cases}$
• D. $k(x) = \begin{cases} x^2, & -\infty < x \le 1
  x, & 1 \le x \le \infty \end{cases}$

Hint:

Always remember that LMVT requires differentiability on the open interval $(a, b)$. Modulus functions are non-differentiable only where their inner expression becomes zero. For $|x-1| + |x-5|$, the non-differentiable points are $x=1$ and $x=5$, which lie exactly on the boundaries, keeping the open interval $(1, 5)$ perfectly differentiable!

Answer 1

The Correct Option is A