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

$(2005,2 M)$

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Answer:

Correct Answer: 16. (a)

Solution:

  1. 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 $x \in(1,3)$



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