Taylor's Theorem Error Term Upper Bound

Question

Use the error term in Taylor's theorem to given an upper bound for $|\cos(x) - f(x)|$ over the interval $[-\pi/2, \pi/2]$, using the worst case of $\varepsilon$, where $f(x)=1-(x^2/2)$.

(A lot of what I see already posted has some degree which I am not provided so I am confused how to solve this).

Edit: Is it fair to say this problem is degree 2 since it is $x^2$?

Answer 1

Let $g(x)= \cos x$. Then , by Taylor, if $ x \in [-\pi/2, \pi/2]:$

$$|\cos(x) - f(x)|= |\frac{g'''(t)}{3!}x^3|$$

for some $t$ between $0$ and $x$.

Can you proceed ?