The minimum value of e^{(2x^2 - 2x + 1)sin^2 | vedclass

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(C) Let $y = e^{(2x^2 - 2x + 1)\sin^2 x}$.Since the exponential function $e^u$ is an increasing function,the minimum value of $y$ occurs when the expo

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$1$

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(C) Let $y = e^{(2x^2 - 2x + 1)\sin^2 x}$.Since the exponential function $e^u$ is an increasing function,the minimum value of $y$ occurs when the exponent $f(x) = (2x^2 - 2x + 1)\sin^2 x$ is minimized.Note that $2x^2 - 2x + 1 = 2(x^2 - x + 1/4) + 1/2 = 2(x - 1/2)^2 + 1/2$,which is always positive (minimum value is $1/2$).Also,$\sin^2 x \ge 0$ for all $x \in \mathbb{R}$.Since both factors are non-negative,the minimum value of $f(x)$ is $0$,which occurs when $\sin^2 x = 0$ (i.e.,$x = n\pi$ for any integer $n$).Therefore,the minimum value of $y$ is $e^0 = 1$.

The minimum value of $e^{(2x^2 - 2x + 1)\sin^2 x}$ is

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