Let f: R R, g: R R and h: R R be differentiable functions such th… - Vidyadip

MCQ

Let $f: \mathbb{R} \rightarrow \mathbb{R}, \quad g: \mathbb{R} \rightarrow \mathbb{R}$ and $h: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that $f(x)=x^3+3 x+2, g(f(x))=x$ and $h(g(g(x)))=x$ for all $x \in \mathbb{R}$. Then

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

A
$ABD$
B
$ABC$
C
$AB$
✓
$BC$

✓

Answer

Correct option: D.

$BC$

d
$g^{\prime}(f(x)) \cdot f^{\prime}(x)=1$

$g^{\prime}(2) \cdot f^{\prime}(0)=1$

$g^{\prime}(2)=\frac{1}{f^{\prime}(0)}$

$f^{\prime}(x)=3 x^2+3$

$g^{\prime}(2)=\frac{1}{3}$

$h(g(g(x))=x$

$h(g(g(f(x)))=f(x)$

$h(g(x))=f(x)$

$h(g(3))=f(3)=38$

$h(g(f(x)))=f(f(x))$

$h(x)=f(f(x))$

$h^{\prime}(x)=f^{\prime}(f(x)) \cdot f^{\prime}(x)$

$h^{\prime}(1)=f^{\prime}(f(1)) \cdot f^{\prime}(1)$

$=111 \times 6=666$

$h(0)=f(f(0))=f(2)=16$

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