First slide
Maxima and minima

If A>0,B>0, and A+B=π3,then the maximum value of tan A tan B is


Given A+B=60 or B=60A
 tan B=tan60A=3tan A1+3tan A
 Now z=tan Atan B
or    z=t(3t)1+3t=3tt21+3t
where t = tan A
or     t=1/3
or     t=tan A=tan 30
The other value is rejected as both A and B are +ve acute angles.
If t<13,dzdtis positive and if t>13,dzdtis negative.
Hence, maximum when t=13and maximum value =13.

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