Let f(x)=3^{(x^{2}-2)^{3}+4}, x in R. Then wh | vedclass

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(D) Given $f(x) = 3^{(x^2-2)^3+4} = 81 \cdot 3^{(x^2-2)^3}$.First derivative: $f'(x) = 81 \cdot 3^{(x^2-2)^3} \cdot \ln 3 \cdot 3(x^2-2)^2 \cdot 2x =

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All,$P, Q$ and $R$

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(D) Given $f(x) = 3^{(x^2-2)^3+4} = 81 \cdot 3^{(x^2-2)^3}$.First derivative: $f'(x) = 81 \cdot 3^{(x^2-2)^3} \cdot \ln 3 \cdot 3(x^2-2)^2 \cdot 2x = (486 \ln 3) \cdot 3^{(x^2-2)^3} \cdot x(x^2-2)^2$.For $P$: $f'(x) = 0$ at $x=0$ and $x=\pm \sqrt{2}$. For $x$ near $0$,$x(x^2-2)^2$ changes sign from negative to positive,so $x=0$ is a point of local minima. Thus,$P$ is true.For $Q$: $f''(x) = 0$ at $x=\sqrt{2}$. Since $f''(x)$ changes sign at $x=\sqrt{2}$ (as $(x^2-2)$ is a factor),$x=\sqrt{2}$ is a point of inflection. Thus,$Q$ is true.For $R$: For $x > \sqrt{2}$,$f'(x) > 0$. Analyzing $f''(x)$,for $x > \sqrt{2}$,$f''(x) > 0$,which implies $f'(x)$ is increasing. Thus,$R$ is true.Therefore,all statements $P, Q,$ and $R$ are true.

Let $f(x)=3^{(x^{2}-2)^{3}+4}, x \in R$. 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 $f$
$R: f^{\prime}$ is increasing for $x>\sqrt{2}$

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Let $f(x)=3^{(x^{2}-2)^{3}+4}, x \in R$. 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^{\prime}$ is increasing for $x>\sqrt{2}$