Limit Continuity and Differentiability 7 Question 16
16. Let $f(x)=|| x|-1|$, then points where, $f(x)$ is not differentiable is/are
(a) $0, \pm 1$
(b) \pm 1
(c) 0
(d) 1
$(2005,2 M)$
Show Answer
Answer:
Correct Answer: 16. (a)
Solution:
- Let, $g(x)=f(x)-x^{2}$
$\Rightarrow g(x)$ has atleast 3 real roots which are $x=1,2,3$
[by mean value theorem]
$\Rightarrow \quad g^{\prime}(x)$ has atleast 2 real roots in $x \in(1,3)$
$\Rightarrow g^{\prime \prime}(x)$ has atleast 1 real roots in $x \in(1,3)$
$\Rightarrow f^{\prime \prime}(x)-2 \cdot 1=0$. for atleast 1 real root in $x \in(1,3)$
$\Rightarrow f^{\prime \prime}(x)=2$, for atleast one root in $x \in(1,3)$
