Prove that tan &#x2061;<!-- ⁡ --> x &lt; 4 &#x03C0;<!-- π --> </mfra

vasorasy8 2022-07-04 Answered
Prove that
tan x < 4 π x , x ( 0 ; π 4 )
You can still ask an expert for help

Want to know more about Trigonometry?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Elias Flores
Answered 2022-07-05 Author has 24 answers
Let g be the function defined on the interval [ 0 , π / 4 ] as
g ( x ) = { tan x x , x 0 1 , x = 0
Then, the derivative g′ of g is given by
g ( x ) = x sec 2 x tan x x 2 = x 1 2 sin ( 2 x ) x 2 cos 2 x > 0
for x>0 and g′(0)=0.
Inasmuch as g is increasing for x [ 0 , π / 4 ] it attains, therefore, its maximum there at x = π / 4
Thus,
g ( x ) < g ( π / 4 ) tan x x < 1 π / 4 tan x < 4 π x
and we are done!
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions