Question

With Taylor/Maclaurin polynomials

Use the Remainder Estimation Theorem to find an interval containing x = 0 over which f(x) can be approximated by p(x) to three decimal-place accuracy throughout the interval.

f(x) = sin x
p(x) = x - 1/6 * x^3

The book answer is (-.569, +.569)
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My work so far

For three decimal-place accuracy
R(x) <= 5 * 10^-4

For MacLaurin of x - 1/6 * x^3, n must equal 4 so
R(x) <= M/24 * x^4

f to the 4th derivative over interval <= M
sin x over interval <= M

R(x) <= sin x / 24 * x^4
5*10^-4 >= sin x / 24 * x^4

Is this right so far? I can use a computer to find several solutions to the above equation but they don't match the book's answer.