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Question

If f(θ)=sin4θ+cos2θ, then range of f(θ) is

A
[12,1]
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B
[12,34]
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C
[34,1]
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D
None of these
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Solution

The correct option is A [34,1]
f(θ)=sin4θ+cos2θ=sin4θ+(1sin2θ)

Put sin2θ=t, we get

f(t)=t2t+1

f(t) is quadratic equation with D<0

minimum value of f(t), when

df(t)dt=0t=12

f(t)min=1412+1=34

and the maximum value of t=1, therefore the maximum value of f(t)

f(t)=11+1=1

range of f(t)[34,1]

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