Find the minimum value of $e^{(2 x - 2 x + 1) {sin}^{2} x}$ .

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Solution

Let $f (x) = (2 x^{2} - 2 x + 1) {sin}^{2} x$
Minimum value of ${sin}^{2} x = 0$
$\Rightarrow x = 0$
$∴ e^{(2 x^{2} - 2 x + 1) {sin}^{2} x} = e^{0} = 1.$
Thus, the minimum value is $1$

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