Insane Trig. Inequality!

Calculus Level 5

Let $x\in\Big [-\frac{5\pi}{12}, -\frac{\pi}{3}\Big ]$ . The maximum value of

$\tan\bigg (x+\frac{2\pi}{3}\bigg )-\tan\bigg (x+\frac{\pi}{6} \bigg )+\cos\bigg (x+\frac{\pi}{6}\bigg )$

can be expressed as $\frac{a}{b}\sqrt{c}$ where $a$ and $b$ are coprime positive integers and $c$ is not a square number. What is the value of $a+b+c$ ?

The answer is 20.

2 solutions

Shriya Mandarapu
May 17, 2014

the equation given is an increasing function in the given interval which can be checked using the first derivative test..then the equation would take its maximum at the right extreme of the interval i.e; -pi/6...substituting this would give 11/6 sqrt{3} where a=11 b=6 c=3 a+b+c=20.

using derivative test the max value will come at interval pi/6 and on substituting it, the ans will come as 11/6sqrt(3) so therefore on comparing a+b+c=20.

Sudipan Mallick - 7 years ago

But it says interval is between -5pi/12 and -pi/3 that is between -75deg n -60 deg so how can we have the maxima at pi/6 (30 deg)......it is not in the range right ??

Pranit Bankar - 7 years ago

Hassan Fahmy
May 28, 2014

the final answer was 11/6√3 so a=11 b=6 c=3 and a+b+c= 11+6+3=20

Nice mockery you are making of yourself

Kumar Krish - 2 years, 2 months ago

Insane Trig. Inequality!

The answer is 20.

2 solutions

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