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  4. The minimum value of e(2x2-2x+1) sin 2x is

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

Rajasthan PETRajasthan PET 2012

Solution:

Given, $ y={{e}^{(2{{x}^{2}}-2x+1){{\sin }^{2}}x}} $
$ 2{{x}^{2}}-2x+1>0 $
for all $ x, $
$ y $ is minimum for $ sin\text{ }x=0 $
$ \Rightarrow $ $ x=0 $
$ \therefore $ $ {{y}_{\min }}={{e}^{0}}=1 $