Given-f-x-ax-2-bx-c-x-and-C-f-its-graph-Determine-the-real-numbers-a-b-and-c-such-that-C-f-passes-through-the-points-A-1-2-B-4-8-and-has-a-tangent-parallel-to-the-x-axis-at-x-2-

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Question Number 97143 by Ar Brandon last updated on 06/Jun/20

$$\mathrm{Given}\mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}}{\mathrm{x}}\mathrm{and}{\mathcal{C}}_{\mathrm{f}}\mathrm{its}\mathrm{graph};$$$$\mathrm{Determine}\mathrm{the}\mathrm{real}\mathrm{numbers}\mathrm{a},\mathrm{b},\mathrm{and}\mathrm{c}\mathrm{such}\mathrm{that}$$$${\mathcal{C}}_{\mathrm{f}}\mathrm{passes}\mathrm{through}\mathrm{the}\mathrm{points}\mathrm{A}(1;2);\mathrm{B}(-4;8)\mathrm{and}\mathrm{has}$$$$\mathrm{a}\mathrm{tangent}\mathrm{parallel}\mathrm{to}\mathrm{the}\mathrm{x}-\mathrm{axis}\mathrm{at}\mathrm{x}=2.$$

Answered by MJS last updated on 06/Jun/20

$$\mathrm{just}\mathrm{solve}\mathrm{the}\mathrm{linear}\mathrm{system}$$$$\{\begin{array}{l}f\left(1\right)=2\\ f(-4)=8\\ f^{\prime }\left(2\right)=0\end{array}$$$$\mathrm{for}a,b,c$$$$\Rightarrow$$$$a=-\frac{3}{5};b=5;c=-\frac{12}{5}$$

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