Trigonometric Crush 6

Geometry Level 3

$\large{\begin{aligned} t_1 = (\tan x)^{\cot x} &\quad,\quad& t_2 = (\cot x)^{\cot x} \\ t_3 = (\tan x)^{\tan x} &\quad,\quad& t_4 = (\cot x)^{\tan x} \end{aligned}}$

Given that $0 < x< \frac\pi4$ , which of these inequalities in the answer choices must be true?

t_2>t_4>t_1>t_3

t_4>t_2>t_1>t_3

t_2>t_3>t_4>t_1

t_2>t_4>t_3>t_1

1 solution

Raj Rajput
Aug 23, 2015

you can take any value of x between 0<x<pi/4 and compare but approach is

Don't you have to prove that it's true for all $x$ in that interval? Not just for small $k_1, k_2$ .

Pi Han Goh - 5 years, 7 months ago

i think as they are strictly increasing and decreasing in 0<x<pi/4, if relation is true for one value in the given interval then it is true for all

RAJ RAJPUT - 5 years, 7 months ago

You need to show that they are increasing/decreasing

Pi Han Goh - 5 years, 7 months ago

@Pi Han Goh – we can know that through graph and now i cannot edit this as it is handwritten

RAJ RAJPUT - 5 years, 7 months ago

@Raj Rajput – There's no point in writing out a solution if you just interpret it by graph. It's like saying "My calculator says so."

Pi Han Goh - 5 years, 7 months ago

@Pi Han Goh – to see whether the function is increasing or decreasing we have to either see its behavior through graph in that particular interval or we can have derivative test both are correct ways but graph way is shortest so i said that we can see it through graph .....

RAJ RAJPUT - 5 years, 7 months ago

@Raj Rajput – You did not show any of these methods in your solution, so your solution is wrong.

Pi Han Goh - 5 years, 7 months ago

@Pi Han Goh – thanks for catching that mistake, next time I will try a level better than this :)

RAJ RAJPUT - 5 years, 7 months ago

Trigonometric Crush 6

1 solution

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