JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Application of Derivatives

Write $f(x)=81\cdot 3^{(x^2-2{)}^3}$. Then:


The sign of $f^{\prime }(x)$ depends on $x$ (since the other factors are always positive). $f^{\prime }(x)=0$ at $x=0,\pm \sqrt{2}$. For $x<0$, $f^{\prime }<0$; for $x>0$, $f^{\prime }>0$. Thus $x=0$ is a local minimum. Statement P is true.

For the second derivative, compute $f^{\prime \prime }(x)$. At $x=\sqrt{2}$, the factor $(x^2-2)$ changes sign, and analysis shows $f^{\prime \prime }(\sqrt{2})=0$ with $f^{\prime \prime }$ changing sign, so $x=\sqrt{2}$ is an inflection point. Statement Q is true.

For $x>\sqrt{2}$, $f^{\prime \prime }(x)>0$, so $f^{\prime }$ is increasing. Statement R is true.

All three statements P, Q, and R are true.