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  • Let <img alt="\mathrm{f(x)=3^{\left(x^{2}-2\right)^{3}+4}, x \in \mathbf{R}.}" src="/latex-image/?%5Cmathrm%7Bf%28x%29%3D3%5E%7B%5Cleft%28x%5E%7B2%7D-2%5Cright%29%5E%7B3%7D&plus;4%7D%2C%20x%20%

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

 

Option: 1

Only P and Q
 


Option: 2

Only P and R
 


Option: 3

Only Q and R
 


Option: 4

All P,Q, and R


Answers (1)

best_answer

\begin{aligned} &\mathrm{ f(x)=81 \cdot 3^{\left(x^{2}-2\right)^{3}} }\\ & \mathrm{f(x)=81 \cdot 3^{\left(x^{2}-2\right)^{3}} \ln 3 \cdot 3\left(x^{2}-2\right)^{2} \cdot 2 x }\\ & \mathrm{=(81 \times 6) 3^{\left(x^{2}-2\right)^{3}} \times\left(x^{2}-2\right)^{2} \ln 3} \\ \end{aligned}

\begin{aligned} &\mathrm{ x=6} \text { is point of local min } \\ & \mathrm{f^{\prime}(x)=\underbrace{(4-6 \cdot \ln 3)}_{k} \underbrace{3^{\left(x^{2}-2\right)^{3}} x\left(x^{2}-2\right)^{2}}_{g(x)}} \\ &\mathrm{ 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\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 }\\ & \mathrm{=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] .} \\ &\\ &\end{aligned}  
 \mathrm{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]

\mathrm{f^{\prime \prime}(x)=k \cdot g^{\prime}(x)}

 \begin{aligned} & \mathrm{f^{\prime \prime}(\sqrt{2})=0, f^{\prime \prime}(\sqrt{2}+)>0, f^{\prime \prime}\left(\sqrt{2}^{-}\right)<0}\\ &\mathrm{x=\sqrt{2}} \text{is point of infiection}\\ & \mathrm{f^{\prime \prime}(x)>0} \text{ for }\mathrm{x>\sqrt{2}}\text{ so }\mathrm{f^{\prime}(x)}\text{ is increasing} \end{aligned}
\begin{aligned} & \mathrm{f^{\prime \prime}(\sqrt{2})=0, f^{\prime \prime}(\sqrt{2}+)>0, f^{\prime \prime}\left(\sqrt{2}^{-}\right)<0}\\ &\mathrm{x=\sqrt{2}} \text{is point of infiection}\\ & \mathrm{f^{\prime \prime}(x)>0} \text{ for }\mathrm{x>\sqrt{2}}\text{ so }\mathrm{f^{\prime}(x)}\text{ is increasing} \end{aligned}

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