Question

what is the minimum value of 2^sin^2X +2^cos^2X .​

Answer 1

Answer:

Let y=2sin2x+2cos2x=2sin2x+21−sin2xy=2sin2⁡x+2cos2⁡x=2sin2⁡x+21−sin2⁡x

(2sin2x)2−y⋅2sin2x+2=0(2sin2⁡x)2−y⋅2sin2⁡x+2=0

which is a Quadratic Equation in 2sin2x2sin2⁡x

So, the discriminant must be ≥0≥0

(y)2≥4⋅2⟹y2≥8(y)2≥4⋅2⟹y2≥8

As y>0,y≥22–√y>0,y≥22

The equality occurs if

2sin2x=2

Answer 2

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