Let-x-5pi-12-pi-3-The-maximum-value-of-y-tan-x-2pi-3-tan-x-pi-6-cos-x-pi-6-is-

Question Number 96758 by bobhans last updated on 04/Jun/20

Let x \in [- \frac{5 π}{12}, - \frac{π}{3}] . The maximum

value of y = \tan (x + \frac{2 π}{3}) - \tan (x + \frac{π}{6}) + \cos (x + \frac{π}{6})

is___

Commented by john santu last updated on 04/Jun/20

set m = - x - \frac{π}{6}, m \in [\frac{π}{6}, \frac{π}{4}]

and 2 m \in [\frac{π}{3}, \frac{π}{2}] . we have

\tan (x + \frac{2 π}{3}) = - \cot (x + \frac{π}{6}) = \cot m

then y = \cot m + \tan m + \cos m

y = \frac{2}{\sin 2 m} + \cos m .

s i n c e b o t h \frac{2}{\sin 2 m} and \cos m a r e

m o n o t o n i c d e c r e a s i n g i n t h i s

c a s e, so y reaches the maksimum

at m = \frac{π}{6}, where y_{\max} = \frac{2}{\sin \frac{π}{3}} + \cos \frac{π}{6}

= \frac{4}{\sqrt{3}} + \frac{\sqrt{3}}{2} = \frac{11 \sqrt{3}}{6} .