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Maxima and Minima — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMaxima and Minima1 Mark
Question
The minimum value of $\text{x}\log_{\text{e}}\text{x}$ is equal to :
  1. $\text{e}$
  2. $\frac{1}{\text{e}}$
  3. $\frac{-1}{\text{e}}$
  4. $\text{2}{\text{e}}$
✓

Answer

  1. $\frac{-1}{\text{e}}$
Solution :
Here, $\text{f}(\text{x})=\text{x}\log_{\text{e}}\text{x}$
lmplies that $\text{f}'(\text{x})=\log_{\text{e}}\text{x}+1$
For a local maxima or a local minima, we must have f'(x) = 0
lmplies that $\log_{\text{e}}\text{x}+1=0$ 
lmplies that $\log_\text{e}\text{x}=-1$
lmplies that $\text{x}=\text{e}^{-1}$
Now, $\text{f}''(\text{x})=\frac{1}{\text{x}}$
lmplies that $\text{f}''(\text{e}^{-1})=\text{e}>0$
Threrfore, $\text{f}(\text{e}^{-1})$ is a local minima.
Hence, the minimum value of $\text{f}(\text{x})=\text{f}(\text{e}^{-1})$
lmplies that $\text{e}^{-1}\log_\text{e}(\text{e}^{-1})=-\text{e}^{-1}=\frac{-1}{\text{e}}$

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