Question

Let $f:R\to R,g:R\to R$ and $h:R\to R$ be differentiable functions such that $f\left(x\right)=x^3+3x+2,g\left(f\right(x\left)\right)=x$ and $h\left(g\right(g\left(x\right)\left)\right)=x$ for all $x\in R$. Then

• A. $h^{\prime }\left(1\right)=666$
• B. $h\left(0\right)=16$
• C. $h\left(g\right(3\left)\right)=36$
• D. $g^{\prime }\left(2\right)=\frac{1}{15}$

Answer 1

Answer: (A), (B)

$h\left(0\right)=16$

$g^{\prime }\left(f\right(x\left)\right)\cdot f^{\prime }\left(x\right)=1$
$g^{\prime }\left(2\right)\cdot f^{\prime }\left(0\right)=1$
$g^{\prime }\left(2\right)=\frac{1}{f^{\prime }\left(0\right)}$
$f^{\prime }\left(x\right)=3x^2+3$
$g^{\prime }\left(2\right)=\frac{1}{3}$
$h\left(g\right(g\left(x\right))=x$
$h\left(g\right(g\left(f\right(x\left)\right))=f(x)$
$h\left(g\right(x\left)\right)=f\left(x\right)$
$h\left(g\right(3\left)\right)=f\left(3\right)=38$
$h\left(g\right(f\left(x\right)\left)\right)=f\left(f\right(x\left)\right)$
$h\left(x\right)=f\left(f\right(x\left)\right)$
$h^{\prime }\left(x\right)=f^{\prime }\left(f\right(x\left)\right)\cdot f^{\prime }\left(x\right)$
$h^{\prime }\left(1\right)=f^{\prime }\left(f\right(1\left)\right)\cdot f^{\prime }\left(1\right)$
$=111\times 6=666$
$h\left(0\right)=f\left(f\right(0\left)\right)=f\left(2\right)=16$