Help with Calculus?

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1 Answer
Noah G
May 9, 2018

Use the taylor series for #cosx#, which is #1 - x^2/(2!) + x^4/(4!) + ...#

We now rewrite the inequality.

#0 ≤ 1- x^2/(2!) + x^4/(4!) - 1 + x^2/(2!) ≤ 1/24#

#0 ≤ x^4/(4!) ≤ 1/24#

The maximum of this will be at #x = 1 or -1#, or #1/(4!) = 1/24#. The minimum will be at #x = 0# or #0#.

If we add more terms it makes no difference.

#0 ≤ x^4/(4!) - x^6/(6!) ≤ 1/24#

The minimum will ALWAYS be #0# no matter the value of #x#. The maximum will be at most #1/24#, because the terms after #x^4/(4!)# converge to #0# when #x= 1#.

We have therefore proved the required statement.

Hopefully this helps!

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