Maximum Error in Taylor Polynomial for cos(x) on Interval [-.25, .25]

[negatifzeo]

Homework Statement

Find the maximum error in approximating cos(x) by its Taylor polynomial of order 2 on the
interval [
−.25, .25]. Justify your answer using the Remainder Estimation Theorem.

Homework Equations

|R3(x)<=M/3! |x|^3

The Attempt at a Solution

|R3(x)<=M/3! |x|^3 Plugging in the 3 is easy enough, but I don't understand where the M comes from. What is M here? I initally thought it might be the value of the 4th taylor polynomial, but that would make the remainder less than or = zero, right?

 

[Wretchosoft]

M is the maximum of the absolute value of the 3rd derivative of cos(x) on [-1/4,1/4].

 

[negatifzeo]

Thank you so much!