QJEEM21Ma0602: JEE Main 2021 Mathematics - Class 12 JEEM21Ma0602 Solution: Let the functions f : R R and g : R R be defined as : f ( x ) = …

Question

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Let the functions f : R

\to

R and g : R

\to

R be defined as :

f (x) = {\begin{matrix} x + 2, & x < 0 \\ x^{2}, & x \geq 0 \end{matrix}

and

g (x) = {\begin{matrix} x^{3}, & x < 1 \\ 3 x - 2, & x \geq 1 \end{matrix}

Then, the number of points in R where (fog) (x) is NOT differentiable is equal to :

Accepted Answer

f o g (x) = {\begin{matrix} x^{3} + 2, & x \leq 0 \\ x^{6}, & 0 \leq x \leq 1 \\ {(3 x - 2)}^{2}, & x \geq 1 \end{matrix}

\because

fog(x) is discontinuous at x = 0 then non-differentiable at x = 0

Now,

at x = 1

RHD = \mathop {\lim }\limits_{h \to 0} {{f(1 + h) - f(1)} \over h} = \mathop {\lim }\limits_{h \to 0} {{{{(3(1 + h) - 2)}^2} - 1} \over h} = 6

LHD = \mathop {\lim }\limits_{h \to 0} {{f(1 - h) - f(1)} \over { - h}} = \mathop {\lim }\limits_{h \to 0} {{{{(1 - h)}^6} - 1} \over { - h}} = 6

Number of points of non-differentiability = 1

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