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The area boun...

Question

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 is:

1 - 3 e^{- 2}

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1 - 2 e^{- 2}

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1 - e^{- 2}

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1

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Solution

The correct option is C $1 - 3 e^{- 2}$
$y^{'} = e^{- x} - x e^{- x} = e^{- x} (1 - x)$
$y "= - e^{- x} - e^{- x} (1 - x) \Rightarrow - e^{- x} (2 - x) = 0 \Rightarrow x = 2$
So point of inflection is x = 2
Hence required area is, A = $\int_{0}^{2} x e^{- x} d x = [- x e^{- x}]_{0}^{2} - e^{- x}]_{0}^{2} = 1 - 3 e^{2}$

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