Question

Help with Calculus?

Answer 1

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!