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If d(x)= xsinx is examined on the interval [0,2pi], what is the maximum value?

Taking the derivative gives xcosx + sinx.
Setting to 0 gives xcosx + sinx = 0.
Dividing by cosx gives x + sinx/cosx = 0.
An identity is sinx/cosx = tanx, so we have x + tanx = 0.
That happens where x = -tanx.

Plotting a chart of xsinx in Excel tells me the maximum is between 1.8 and 2.2.
Using the bisection method gives me the following list, with x1, f(x1), x2, f(x2):
1.8000   -2.486261675   2.2000    0.826176943
2.0000   -0.185039863   2.2000    0.826176943
2.0000   -0.185039863   2.1000    0.390153457
2.0000   -0.185039863   2.0500    0.125439328
2.0250   -0.023130187   2.0500    0.125439328
2.0250   -0.023130187   2.0375    0.05268754
2.0250   -0.023130187   2.0312    0.015177598
2.0281   -0.003874511   2.0312    0.015177598
2.0281   -0.003874511   2.0296    0.005676729
2.0281   -0.003874511   2.0289    0.000907438

This means the answer is between 2.0281 and 2.0289.
From the values of f(x), the answer is probably near 2.0288,
but I'm not sure if this needs to be that accurate.  


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