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Show that The maximum value of sinx. cos...

Show that The maximum value of sinx. cosx in R is `1/2`

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We have, `f(x)=sinx.cosx=1/2sin2x`
`therefore f^(')(x)=1/2.cos2x.2=cos2x`
Now, `f^(')(x)=0 rArr cos2x=0`
`rArr cos2x=cospi/2 rArr x=pi/4`
Also, `f^('')(x)_("at x"=pi//4) =-2 sin 2.pi/4=-2sinpi/2=-2lt0`
At `pi/4`, f(x) is maximum and `pi/4` is point of maxima.
`therefore f(pi/4)=1/2.sin2.pi/4=1/2`
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