Question

Two functions $f$ & $g$ have first & second derivatives at $x=0$ & satisfy the relations, $f\left(0\right)=\frac{2}{g\left(0\right)},f^{\prime }\left(0\right)=2g^{\prime }\left(0\right)=4g\left(0\right),g^{\prime \prime }\left(0\right)=5f^{\prime \prime }\left(0\right)=6f\left(0\right)=3$ then

• A. if $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ then $h^{\prime }\left(0\right)=\frac{15}{4}$
• B. if $k\left(x\right)=f\left(x\right)\cdot g\left(x\right)\sin x$ then $k^{\prime }\left(0\right)=2$
• C. $\underset{x\to 0}{lim}\frac{g^{\prime }\left(x\right)}{f^{\prime }\left(x\right)}=\frac{1}{2}$
• D. None of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A if $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ then $h^{\prime }\left(0\right)=\frac{15}{4}$
B if $k\left(x\right)=f\left(x\right)\cdot g\left(x\right)\sin x$ then $k^{\prime }\left(0\right)=2$
C $\underset{x\to 0}{lim}\frac{g^{\prime }\left(x\right)}{f^{\prime }\left(x\right)}=\frac{1}{2}$

$A:h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}\therefore h^{\prime }\left(x\right)=-\frac{f\left(x\right)}{{\left(g\right(x\left)\right)}^2}{g}^{\prime }\left(x\right)+\frac{f^{\prime }\left(x\right)}{g\left(x\right)}\therefore h^{\prime }\left(0\right)=-\frac{f\left(0\right)}{{\left(g\right(0\left)\right)}^2}{g}^{\prime }\left(0\right)+\frac{f^{\prime }\left(0\right)}{g\left(0\right)}=-\frac{1/2}{{\left(4\right)}^2}\left(8\right)+\frac{16}{4}=-1/4+4=15/4B:k\left(x\right)=f\left(x\right)g\left(x\right)sin\left(x\right)\therefore k^{\prime }\left(x\right)=f\left(x\right)g\left(x\right)cos\left(x\right)+f^{\prime }\left(x\right)g\left(x\right)sin\left(x\right)+f\left(x\right)g^{\prime }\left(x\right)sin\left(x\right)\therefore k^{\prime }\left(0\right)=f\left(0\right)g\left(0\right)cos\left(0\right)+f^{\prime }\left(0\right)g\left(0\right)sin\left(0\right)+f\left(0\right)g^{\prime }\left(0\right)sin\left(0\right)=f\left(0\right)g\left(0\right)\left(1\right)+0+0=\frac{1}{2}\left(4\right)=2C:\underset{x\to 0}{lim}\frac{g^{\prime }\left(x\right)}{f^{\prime }\left(x\right)}=\frac{g^{\prime }\left(0\right)}{f^{\prime }\left(0\right)}=\frac{8}{16}=\frac{1}{2}$