QJEEM19Madfaa: JEE Main 2019 Mathematics - Class 12 JEEM19Madfaa Solution: Let f : R R be differentiable at c R and f(c) = 0. If g(x) =…

Question

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Let f : R

\to

R be differentiable at c

\in

R and f(c) = 0. If g(x) = |f(x)| , then at x = c, g is :

Accepted Answer

g'(c) = \mathop {\lim }\limits_{x \to c} {{g(x) - g(c)} \over {x - c}}

\Rightarrow g'(c) = \mathop {\lim }\limits_{x \to c} {{\left| {f(x)} \right| - \left| {f(c)} \right|} \over {x - c}}

\therefore

f(c) = 0

\Rightarrow g'(c) = \mathop {\lim }\limits_{x \to c} {{\left| {f(x)} \right|} \over {x - c}}

\Rightarrow g'(c) = \mathop {\lim }\limits_{x \to c} {{f(x)} \over {x - c}}

if f(x) > 0

and

g'(c) = \mathop {\lim }\limits_{x \to c} {{ - f(x)} \over {x - c}}

if f(x) < 0

\Rightarrow g'(c) = f'(c) = - f'(c)

\Rightarrow

2f'(c) = 0

\Rightarrow

f'(c) = 0

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