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Question

Consider the function y=f(x)=ln(1+sinx) with 2πx2π.

Find local maxima and minima of f(x)

A
maxima at x=π2 and no minima,
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B
maxima at 3π2 and no minima,
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C
maxima at 3π2 and minima at x=π2
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D
maxima at x=π2 and minima at 3π2
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Solution

The correct options are
A maxima at x=π2 and no minima,
B maxima at 3π2 and no minima,
y=ln(1+sinx)

dydx=cosx1+sinx=0 for extremum.

x=±π2,±3π2

But at x=π2,3π2 ln(1+sinx) is not defined

Thus x=π2,3π2

d2ydx2=(1+sinx)sinxcosx.cosx(1+sinx)2=11+sinx<0x the domain of f.

Hence both x=π2, and x=3π2 are point of local maxima.

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