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=60∘−A∴ tan ⁡B=tan⁡60∘−A=3−tan ⁡A1+3tan ⁡A Now z=tan⁡ Atan⁡ Bor    z=t(3−t)1+3t=3t−t21+3twhere t = tan A       dzdt=−(t+3)(3t−1)(1+3t)2=0or     t=1/3or     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.