Error bounds and accuracy pitfalls for an approximate root

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Problem 1.23.39 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"
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(1)

Give an example of an equation f(x)=0f(x) = 0 with a root ξ\xi and its approximate value x∗x^{*} such that the inequalities

f′(x)>0 and ∣x∗−ξ∣⩾103,∣f(x∗)∣⩽10−3, f^{\prime }(x) > 0 \text{ and }\left|x^{*}-\xi \right| \geqslant 10^{3}, \quad \left|f\left(x^{*}\right)\right| \leqslant 10^{-3},

i.e., verify that from the "closeness" of f(x∗)f\left(x^{*}\right) to zero it does not follow that x∗x^{*} is "close" to the root ξ\xi;

(2)

give an example of an equation f(x)=0f(x) = 0 with a root ξ\xi and its approximate value x∗x^{*} such that the inequalities

f′(x)>0 and ∣x∗−ξ∣⩽10−3,∣f(x∗)∣⩾103, f^{\prime }(x) > 0 \text{ and }\left|x^{*}-\xi \right| \leqslant 10^{-3}, \quad \left|f\left(x^{*}\right)\right| \geqslant 10^{3},

i.e., verify that from the "closeness" of the approximate value x∗x^{*} to the root ξ\xi it does not follow that f(x∗)f\left(x^{*}\right) is "close" to zero.

Problem 1.23.66 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

Prove that the smallest positive root ξ\xi of the equation ctg⁡x=λ−x,λ>2\operatorname {ctg} x = \lambda -x, \quad \lambda > 2, satisfies the inequality

ξ<2λ+λ2−4 \xi < \frac{2}{\lambda +\sqrt{\lambda ^{2}-4}}
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References

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.