Let the functions f : R $\to$ R and g : R $\to$ R be defined as :

$$f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} & {x \ge 0} \cr } } \right.$$ and

$$g(x) = \left\{ {\matrix{ {{x^3},} & {x < 1} \cr {3x - 2,} & {x \ge 1} \cr } } \right.$$

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

$$fog(x) = \left\{ {\matrix{ {{x^3} + 2,} &amp; {x \le 0} \cr {{x^6},} &amp; {0 \le x \le 1} \cr {{{(3x - 2)}^2},} &amp; {x \ge 1} \cr } } \right.$$<br><br>$\because$ fog(x) is discontinuous at x = 0 then non-differentiable at x = 0<br><br>Now, <br><br>at x = 1<br><br>$$RHD = \mathop {\lim }\limits_{h \to 0} {{f(1 + h) - f(1)} \over h} = \mathop {\lim }\limits_{h \to 0} {{{{(3(1 + h) - 2)}^2} - 1} \over h} = 6$$<br><br>$$LHD = \mathop {\lim }\limits_{h \to 0} {{f(1 - h) - f(1)} \over { - h}} = \mathop {\lim }\limits_{h \to 0} {{{{(1 - h)}^6} - 1} \over { - h}} = 6$$<br><br>Number of points of non-differentiability = 1