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Consider-f-x-x-3-2x-1-Use-the-intermidiate-value-theorem-and-the-Rolle-theorem-to-establish-that-the-equation-f-x-0-has-a-unique-solution-denoted-a-0-0-1-




Question Number 159121 by physicstutes last updated on 13/Nov/21
Consider  f(x) = x^3  + 2x −1.  Use the intermidiate value theorem and  the Rolle theorem to establish that the  equation f(x) = 0 has a unique solution  denoted a_0 ∈] 0,1[.
$$\mathrm{Consider} \\ $$$${f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{2}{x}\:−\mathrm{1}. \\ $$$$\mathrm{Use}\:\mathrm{the}\:\mathrm{intermidiate}\:\mathrm{value}\:\mathrm{theorem}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{Rolle}\:\mathrm{theorem}\:\mathrm{to}\:\mathrm{establish}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution} \\ $$$$\left.\mathrm{denoted}\:{a}_{\mathrm{0}} \in\right]\:\mathrm{0},\mathrm{1}\left[.\:\right. \\ $$

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