Maxima and Minima — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsMaxima and Minima3 Marks
Question
Show that $\frac{\text{logx}}{\text{x}}$ has a minimum value at x = e.
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Answer
Here, $\text{f}(\text{x})=\frac{\log\text{x}}{\text{x}}$ $\Rightarrow\text{f}'(\text{x})=\frac{1-\log\text{x}}{\text{x}^{2}}$ For the local maxima or minima, We have f'(x) = 0 $\Rightarrow\frac{1-\log\text{x}}{\text{x}^{2}}=0$ $\Rightarrow1=\log\text{x}$ $\Rightarrow\log\text{e}=\log\text{x}$ $\Rightarrow\text{x}=\text{e}$ Now, $\text{f}''(\text{x})=\frac{\text{x}^{2}(\frac{-1}{\text{x}})-2\text{x}(1-\log\text{x})}{\text{x}^{4}}=\frac{-3+2\log\text{x}}{\text{x}^{3}}$ $\Rightarrow\text{f}''(\text{e})=\frac{-3+2\log\text{x}}{\text{x}^{3}}=\frac{-1}{\text{e}^{3}}<0$ So, x = e is the point of local maximum.
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