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Q.
Let $f: R \rightarrow R$ be defined as $f(x)=x^3+3 x+2$ and $g(x)$ be a function such that $g(x)=x f^{-1}(x)-\int\limits_0^{f^{-1}(x)} f(t) d t$, then
Integrals
Solution:
Clearly, $f ( x )$ is bijective,
$\therefore$ invertible
$\Theta g(x)=x f^{-1}(x)-\int\limits_0^{f^{-1}(x)} f(t) d t$
Differentiation both sides
$ g ^{\prime}( x )=1 \cdot f ^{-1}( x )+ x \left( f ^{-1}( x )\right)^{\prime}- f \left( f ^{-1}( x )\right) \cdot\left( f ^{-1}( x )\right)^{\prime} $
$\Rightarrow g ^{\prime}( x )= f ^{-1}( x ) $
$\Rightarrow g ^{\prime}(6)= f ^{-1}(6)=1$
$\text { and } g ^{\prime \prime}( x )=\left( f ^{-1}\right)^{\prime}( x ) \Rightarrow g ^{\prime \prime}(6)=\left( f ^{-1}\right)^{\prime}(6)=\frac{1}{ f ^{\prime}(1)} $
$\Rightarrow g ^{\prime \prime}(6)=\frac{1}{6} $