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We can differentiate  y wrt x and check its value is positive.

\frac{dy}{dx} = x^2(-e^{-x})+e^{-x}*2x\\\\=e^{-x}*x(2-x)\\

dy/dx is positive when x(2-x) is positive as e^{-x} is always positive.

x (2 - x) is positive if   0 < x < 2 .

Hence y is increasing  in the interval  (0, 2)

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