JEE Main 2022 · 29 Jul · Shift 1 · Mathematics Q20

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MathematicsSingle correct+4 / -1Question 20 of 90

Let $f(x)=3^{(x^2-2)^3}+4$ , $x\in\mathbf R$ . Then which of the following statements are true ?

P : $x=0$ is a point of local minima of $f$

Q : $x=\sqrt2$ is a point of inflection of $f$

R : $f'$ is increasing for $x>\sqrt2$

Only P and Q

Only P and R

Only Q and R

All P, Q and R

Answer and worked solution

Official answer: D

Derivative. With $g=(x^2-2)^3$ , $f'=3^g\ln3\cdot6x(x^2-2)^2$ : it changes from negative to positive at 0, so P holds.

Second derivative. $f''=3^g\ln3\,(\ln3\,g'^2+g'')$ with $g''=6(x^2-2)(5x^2-2)$ . Near $\sqrt2$ the $g''$ term dominates and changes sign, so Q holds; for $x>\sqrt2$ both terms are positive, so R holds.

(D)

Topics:Maxima and minima Monotonicity

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JEE Main 2022 · 29 Jul · Shift 1, Mathematics question 20