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Question

Minimum value of 3sinθ+4cosθ is

A
1
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B
3
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C
5
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D
5
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Solution

The correct option is D 5
Let,
f(θ)=3sinθ+4cosθ.
Now, f(θ)=3cosθ4sinθ........(1)
Again f′′(θ)=3sinθ4cosθ........(2)
Now for maximum of minimum value of f(θ) we must have,
f(θ)=0
or, 3cosθ4sinθ=0
or, sinθ3=cosθ4=±sin2θ+cos2θ32+42
or, sinθ=±35, cosθ=±45
Now it is evident from (2) that for sinθ=35, cosθ=45 we have f′′(θ)>0.
In this case we will get minimum value of f(θ).
Now minimum value of f(θ)=32+425=5.

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