Find the area bounded by the curve $y = x e^{- x}; x y = 0$ and x=c where c is the x-coordinate of the curve's inflection point:

1 - 3 e^{- 2}

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1 - e^{- 2}

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1 - 3 e^{- 3}

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1 - 3 e^{- 4}

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Solution

The correct option is A $1 - 3 e^{- 2}$
$y = x e^{- x}$
$y^{'} = e^{- x} - x e^{- x}$
$= e^{- x} (1 - x)$
$y "= - e^{- x} + x e^{- x} - e^{- x}$
$= e^{- x} (x - 2)$
$= 0$
Hence
$x = 2$ is the point of inflection.
Now $x y = 0$ implies $x = 0$ and $y = 0$ . Hence x and y axis.
Now
$\int_{0}^{2} x e^{- x} . d x$
$= [- x e^{- x} - e^{- x}]_{0}^{2}$
$= - [x e^{- x} + e^{- x}]_{0}^{2}$
$= - [3 e^{- 2} - 1]$
$= 1 - 3 e^{- 2}$ .

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