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Question

Let f(x)=sinx+cosx+tanx+sin1x+cos1x+tan1x. If 'M' and 'm' are maximum and minimum values of f(x), then their arithmetic mean of them is equal to?

A
π2+cos1
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B
π2+sin1
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C
π4+tan1+cos1
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D
π4+tan1+sin1
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Solution

The correct option is A π2+cos1
Given,
f(x)=sinx+cosx+tanx+sin1x+cos1x+tan1x
x

f(x)=sinx+cosx+tanx+π2+tam1x

Differentiate with respect to x
f(x)=cosxsinx+sec2x+11+x2
F'(x) is always increasing sinc f'(x)>0

The ranfe of f(x) is given by f(1) and f(-1)
f(1)=sin1+cos1tan1π2+ππ4
=π4+cos1sin1tan1

f(1)=sin1+cos1+tan1+π2+π4
=3π4+sin1+cos1+tan1

A.M.=m+M2
=π4+cos1sin1tan1+3π4+cos1+sin1+tan12

=π+2cos12

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