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Question

If f(x)=2x+cot1 x+log (1+x2x), then f(x)

A
increases in (0, )
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B
decreases in (0, )
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C
neither increases nor decreases in (0, )
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D
increases in (, )
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Solution

The correct option is D increases in (, )
f(x)=2x+cot1 x+log (1+x2x)f'(x)=211+x2+11+x2x(x1+x21)f'(x)=1+2x21+x211+x2f'(x)=1+2x21+x21+x21+x2f'(x)=x2+1+x2(1+x21)1+x2Hence, f'(x)>0 for all x.Hence, f(x) is an increasing function in(,) and, in particular f(x) is an increasing function in (0,).

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