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Question

If ABC is an acute angle triangle, then the minimum value of tanA+tanB+tanC is

A
22
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B
32
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C
33
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D
23
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Solution

The correct option is C 33
In a ABC, we know that
tanA+tanB+tanC=tanAtanBtanC (1)

As tanA,tanB,tanC>0, so using
A.M.G.M., we get
tanA+tanB+tanC3(tanAtanBtanC)1/3
From equation (1),
tanA+tanB+tanC3(tanA+tanB+tanC)1/3(tanA+tanB+tanC)2/33tanA+tanB+tanC33/2tanA+tanB+tanC33

Hence, the minimum value of the given expression is 33.

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