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  4. Let f(x)=3(x2-2)3+4, x ∈ R. Then which of the following statements are true ? P: x =0 is a point of local minima of f Q: x=√2 is a point of inflection of f R: f ' is increasing for x>√2

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Q. Let $f(x)=3^{\left(x^2-2\right)^3+4}, x \in R$. Then which of the following statements are true ? $P : x =0$ is a point of local minima of $f$ $Q: x=\sqrt{2}$ is a point of inflection of $f$ $R$ : $f$ ' is increasing for $x>\sqrt{2}$

JEE MainJEE Main 2022Application of Derivatives

Solution:

$ f ( x )=81 \cdot 3^{\left( x ^2-2\right)^3} $
$f ^{\prime}( x )=81 \cdot 3^{\left( x ^2-2\right)^3} \cdot \ln 3 \cdot 3\left( x ^2-2\right)^2 \cdot 2 x$
$ =(81 \times 6) 3^{\left( x ^2-2\right)^3} \times\left( x ^2-2\right)^2 \ln 3$
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$f ^{\prime}( x )=\underbrace{(486 \cdot \ln 3)}_{ k } \underbrace{3^{\left( x ^2-2\right)^3} x\left( x ^2-2\right)^2}_{ g ( x )}$
$g ^{\prime}( x )=3^{\left( x ^2-2\right)^3}\left( x ^2-2\right)^2+ x \cdot 3^{\left( x ^2-2\right)^3} \cdot 4 x \cdot\left( x ^2-2\right) $
$ + x \cdot\left( x ^2-2\right)^2 \cdot 3^{\left( x ^2-2\right)^3} \ln 3 \cdot 3\left( x ^2-2\right)^2 \cdot 2 x$
$ =3^{\left(x^2-2\right)^3}\left( x ^2-2\right)\left[ x ^2-2+4 x ^2+6 x ^2 \ln 3\left( x ^2-2\right)^3\right] $
$ g ^{\prime}( x )=3^{\left(x^2-2\right)^3}\left( x ^2-2\right)\left[5 x ^2-2+6 x ^2 \ln 3\left( x ^2-2\right)^3\right] $
$ f ^{\prime \prime}( x )= k \cdot g ^{\prime}( x )$
$f ^{\prime \prime}(\sqrt{2})=0, f ^{\prime \prime}\left(\sqrt{2}+>0, f ^{\prime}(\sqrt{2})<0\right. $
$ x^{-}=\sqrt{2} \text { is point of inflection }$
$ f ^{\prime \prime}( x ) >0 \text { for } x >\sqrt{2} \text { so } f ^{\prime}( x ) \text { is increasing }$