QJEEM22Ma7619: JEE Main 2022 Mathematics - Class 12 JEEM22Ma7619 Solution: Let . Then which of the following statements are true? is a…

$f(x) = {3^{{{({x^2} - 2)}^3} + 4}},\,x \in R$

$f(x) = {81.3^{{{({x^2} - 2)}^3}}}$

$f'(x) = {81.3^{{{({x^2} - 2)}^3}}}\ln 2.3({x^2} - 2)2x$

$= (486\ln 2)\left( {{3^{{{({x^2} - 2)}^3}}}({x^2} - 2)x} \right)$

$\Rightarrow x = 0$ is the local minima.

$f''(x) = (486\ln 2)\left( {{3^{{{({x^2} - 2)}^3}}}\,.\,({x^2} - 2)(5{x^2} - 2 + 6{x^2}\ln 3({x^2} - 2))} \right)$

$f''(x) = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = \sqrt 2$

f''\left( {{{\sqrt 2 }^ + }} \right) > 0

f''\left( {{{\sqrt 2 }^ - }} \right) < 0

$\Rightarrow x = \sqrt 2$ is point of inflection

f''(x) > 0\,\forall x > \sqrt 2

$\Rightarrow f'(x)$ is increasing for x > \sqrt 2