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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Let $f, g: R \rightarrow R$ be two real valued functions defined as $f(x)=\left\{\begin{array}{cl}-|x+3| & , \quad x<0 \\ e^{x} & , \quad x \geq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{ll}x^{2}+k_{1} x & , \quad x<0 \\ 4 x+k_{2} & , \quad x \geq 0\end{array}\right.$, where $k_{1}$ and $k_{2}$ are real constants. If $(gof)$ is differentiable at $x=0$, then $(gof) (-4)+(gof)\, (4)$ is equal to
  • A
    $4\left(e^{4}+1\right)$
  • B
    $2\left(2 e ^{4}+1\right)$
  • C
    $4 e ^{4}$
  • ✓
    $2\left(2 e ^{4}-1\right)$
✓

Answer

Correct option: D.
$2\left(2 e ^{4}-1\right)$
d
$f(x)=\left\{\begin{array}{lll}x+3 & ; & x<-3 \\ -(x+3) & ; & -3 \leq x<0 \\ e^{x} & ; & x \geq 0\end{array}\right\}$

$g(x)=\left\{\begin{array}{lll}x^{2}+k_{1} x & ; & x<0 \\ 4 x+k_{2} & ; & x \geq 0\end{array}\right\}$

$g(f(x))=\left\{\begin{array}{lll}f(x)^{2}+k_{1} f(x) & ; & f(x)<0 \\ 4 f(x)+k_{2} & ; & f(x) \geq 0\end{array}\right\}$

$g(f(x))=\left\{\begin{array}{ccc}(x+3)^{2}+k_{1}(x+3) & ; & x<-3 \\ (x+3)^{2}-k_{1}(x+3) & ; & -3 \leq x<0 \\ 4 e^{x}+k_{2} & ; & x>0\end{array}\right\}$

check continuity at $x=0$

${gof}$ $(0)= g \left( f \left(0^{-}\right)\right)= g \left( f \left(0^{+}\right)\right)$

$4+ k _{2}=9-3 k _{1}=4+ k _{2}$

$3 k _{1}+ k _{2}=5$

$( g ( f ( x )))^{\prime}=\left\{\begin{array}{ccc}2( x +3)+ k _{1} & ; & x <-3 \\ 2( x +3)- k _{1} & ; & -3 \leq x <0 \\ 4 e ^{ x } & ; & x \geq 0\end{array}\right\}$

$6- k _{1}=4$

$k _{1}=2$

$\therefore k _{1}=2, k _{2}=-1$

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