If F(x)=f(x)g(x)h(x) for all real x, where f(x),g(x)and hx are differentiable functions at some point x0If  F|(x0)=21F(x0),f|(x0)=4f(x0),g|(x0)=−7g(x0)  and  h'(x0)=kh(x0), then k=

# If $F\left(x\right)=f\left(x\right)g\left(x\right)h\left(x\right)$ for all real $x$, where $f\left(x\right),g\left(x\right)$and $h\left(x\right)$ are differentiable functions at some point ${x}_{0}$If  ${F}^{|}\left({x}_{0}\right)=21F\left({x}_{0}\right),{f}^{|}\left({x}_{0}\right)=4f\left({x}_{0}\right),{g}^{|}\left({x}_{0}\right)=-7g\left({x}_{0}\right)$  and  $h\text{'}\left({x}_{0}\right)=kh\left({x}_{0}\right)$, then $k=$

1. A

22

2. B

26

3. C

28

4. D

24

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### Solution:

$F\left(x\right)=f\left(x\right)g\left(x\right)h\left(x\right)\text{\hspace{0.17em}\hspace{0.17em}}\forall x\in R$

${F}^{|}\left(x\right)={f}^{1}\left(x\right)g\left(x\right)h\left(x\right)+f\left(x\right){g}^{|}\left(x\right)h\left(x\right)+f\left(x\right)g\left(x\right){h}^{|}\left(x\right)$

$\text{at\hspace{0.17em}\hspace{0.17em}}x={x}_{0}$

${F}^{|}\left({x}_{0}\right)={f}^{1}\left({x}_{0}\right)g\left({x}_{0}\right)h\left({x}_{0}\right)+f\left({x}_{0}\right){g}^{|}\left({x}_{0}\right)h\left({x}_{0}\right)+f\left({x}_{0}\right)g\left({x}_{0}\right){h}^{|}\left({x}_{0}\right)$

$21F\left({x}_{0}\right)=4f\left({x}_{0}\right).g\left({x}_{0}\right)h\left({x}_{0}\right)-7f\left({x}_{0}\right)g\left({x}_{0}\right)h\left({x}_{0}\right)+k.h\left({x}_{0}\right)f\left({x}_{0}\right)g\left({x}_{0}\right)$

$21=4-7+k\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left[\because F\left({x}_{0}\right)=f\left({x}_{0}\right).g\left({x}_{0}\right)h\left({x}_{0}\right)\right]$

$⇒k=24$

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