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STD 12 - 5. continuity and differentiation — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 5. continuity and differentiation1 Mark
MCQ
Let the functions $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{ll}x+2, & x<0 \\ x^{2}, & x \geq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{lr}x^{3}, & x<1 \\ 3 x-2, & x \geq 1\end{array}\right.$

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

  • A
    $3$
  • ✓
    $1$
  • C
    $0$
  • D
    $2$
✓

Answer

Correct option: B.
$1$
b
$f(g(x))=\left\{\begin{array}{ll}g(x)+2, & g(x)<0 \\ (g(x))^{2}, & g(x) \geq 0\end{array}\right.$

$=\left\{\begin{array}{ll}x^{3}+2, & x<0 \\ x^{6}, & x \in[0,1) \\ (3 x-2)^{2}, & x \in[1, \infty)\end{array}\right.$

$(f og(x))^{\prime}=\left\{\begin{array}{ll}3 x^{2}, & x<0 \\ 6 x^{5}, & x \in(0,1) \\ 2(3 x-2) \times 3, & x \in(1, \infty)\end{array}\right.$

At ' $O ^{\prime}$

$L.H.L.$ $\neq$ $R.H.L.$ (Discontinuous)

At $'1'$

$L.H.D.$ $=6=$ R.H.D. $\Rightarrow f \operatorname{og}( x )$ is differentiable for $x \in R -\{0\}$

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