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> Let $f: \mathbb{R} \rightarrow \mathbb{R}, f(x):=\cos(x)e^x$.
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> Find a minimal $N \in \mathbb{N}$ such that $$|f(x)-T\_Nf(x,0)| \le 10^{-6} \quad (x \in [-\frac{1}{10}, \frac{1}{10}]).$$
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I've encountered this question here on Math SE before ([here](https://math.stackexchange.com/questions/3509889/how-to-show-leftfx-t-n-fx-0-right-leq-10-6)) but I would like to know how to solve it without using Lagrange.

After identifying all the derivatives up to $f^{(4)}$ (for instance), I got

$$\left|\,\cos(x)e^x-(-\frac16 x^4-\frac13x^3+1)\,\right| = \left|\,\cos(x)e^x+\frac16 x^4+\frac13x^3-1)\,\right| \le \\ \cos(\frac{1}{10})e^{1/10}+ \frac16 \cdot \frac{1}{10^4} + \frac13 \cdot \frac{1}{10^3} -1 \approx 0.11,$$

which isn't even close. So how to approach it?

Thanks in advance!