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Maxima and Minima — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMaxima and Minima5 Marks
Question
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection, $f(x) = xe^x$
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Answer

We have, $\text{f}(\text{x})=\text{x}\text{e}^{\text{x}}$
$\therefore \text{f }(\text{x})=\text{e}^{\text{x}}+\text{xe}^{\text{x}} =\text{e}^{\text{x}}(\text{x}+1)$
$ \text{f}\ ''(\text{x})=\text{e}^{\text{x}}(\text{x}+1)+\text{e}^\text{x}$
$=\text{e}^\text{x}(\text{x}+2)$
For maxima and minima,
$f\ '(x) = 0$
$\Rightarrow\text{e}^{\text{x}}(\text{x}+1)=0$
$\Rightarrow x = -1$
Now, $\text{f}\ ''(-1) =\text{e}-1 =\frac{1}{\text{e}}>0$
$\therefore \text{x} =-1$ is point of local minima.
Hence, local minimum value $= \text{f}(-1) = \frac{-1}{\text{e}}$

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