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Let $f(x)=(1+x^2)^{1/3}$.

Find the second degree Taylor polynomial $T_2(x)$ of $f(x)$ around $x_0=0$

Furthermore, determine a constant $C>0$ such that:

$$ |f(x)-T_2(x)|\leq C|x^3|$$

for all $x \in [-1,1]$


So I have found $T_2(x)$ as such:

$$T_2(x)=\sum_{k=0}^{2} \frac{f^{(k)}x_0}{k!} (x-x_0)^k = 1 + \frac{1}{3}x^2$$

But I don't understand exactly how to determine the constant C. Clearly $ |f(x)-T_2(x)|$ must be the the second degree remainder. I also thought about the range of $f(x)$ and $T_2(x)$ but I don't think that helps in any way. Can someone point me in the right direction?

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Hint: Taylor's Theorem with Remainder term has a very specific formula for the remainder term as follows: for a $3$-times continuously differentiable (weaker conditions work but this does the job) real function $f(x)$ on an interval containing $x_0$, we get $$|R_2(x)| = |f(x) - T_2(x)| = |\frac{f^{(3)}(\xi)}{3!}(x - x_0)^3|$$ for some $\xi \in [x_0, x]$ where $T_2(x)$ is your second degree polynomial. So, in your case, $$|f(x) - T_2(x)| = \frac{|f^{(3)}(\xi)|}{3!}|x|^3 \leq \max\limits_{s \in [-1, 1]}|f^{(3)}(s)|\frac{|x|^3}{3!}$$ I leave it to you to find the third derivative of $f(x)$ and find an upper bound on $\max\limits_{s \in [-1, 1]}|f^{(3)}(s)|$ (you don't need to compute the maximum exactly; just an upper bound on it will do) to get your full constant. The maximum is guaranteed to exist because $f^{(3)}(s)$ is a continuous function on the compact domain $[1, -1]$.

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  • $\begingroup$ Thank you, this makes a lot of sense! $\endgroup$ – CruZ Apr 7 at 19:04
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Hint:

You can use that the Taylor series of $\;(1+u)^{1/3} $ is an alternating series. By the binomial formula, $$(1+u)^{1/3}=1+\tfrac13u-\tfrac 19 u^2+\tfrac 5{81}u^3-\dotsm,$$ hence, by substitution, $$(1+x^2)^{1/3}=1+\tfrac13 x^2-\tfrac 19 x^4 +\tfrac 5{81}x^6-\dotsm,$$ is also an alternating series. Leibniz' test stipulates that, when approximating an convergent alternating series by its sum up to rank $n$, the absolute value of the error is no more than the absolute value of the term of rank $n+1$.

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