CORRECT OPTION?​

CORRECT OPTION?​ Question 33 of 90 Let f be the function given byf(x) 2 x e2x. Find the absolute minimum value of f. 1/2

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=2{\mathrm{xe}}^{2\mathrm{x}} \\ \mathrm{Differentiate} \mathrm{with} \mathrm{respect} \mathrm{to} \mathrm{x} \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=2\left[2{\mathrm{xe}}^{2\mathrm{x}}+{\mathrm{e}}^{2\mathrm{x}}\right] \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=2{\mathrm{e}}^{2\mathrm{x}}\left[2\mathrm{x}+1\right] \\ \mathrm{For} \mathrm{maxima} \mathrm{or} \mathrm{minima}, \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=0 \\ \Rightarrow 2{\mathrm{e}}^{2\mathrm{x}}\left[2\mathrm{x}+1\right]=0 \\ \Rightarrow 2\mathrm{x}+1=0 \\ \Rightarrow \mathrm{x}=-\frac{1}{2} \\ \mathrm{Hence} \\ \mathrm{f}\left(-\frac{1}{2}\right)=2\left(-\frac{1}{2}\right){\mathrm{e}}^{2\times \left(-\frac{1}{2}\right)}=-\frac{1}{\mathrm{e}} \mathrm{which} \mathrm{will} \mathrm{be} \mathrm{minimum} \mathrm{value} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right) \end{gathered}$$

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