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Question Number 143606 by bobhans last updated on 16/Jun/21

Answered by TheHoneyCat last updated on 16/Jun/21

$$f\mathrm{surjective}\iff [1,+\mathrm{\infty }[\subset f\left(\mathbb{R}\right)$$$$$$$$\mathrm{knowing}\mathrm{that}f\in {\mathcal{C}}^0(\mathbb{R},[1,+\mathrm{\infty }\left[\right)$$$$\mathrm{and}\mathrm{that}f\left(x\right)\underset{x\to \mp \mathrm{\infty }}{\to }+\mathrm{\infty }$$$$\mathrm{\forall }y\in \mathbb{R}\mathrm{\exists }x\mid f\left(x\right)=y\Rightarrow [y,+\mathrm{\infty }[\subset f\left(\mathbb{R}\right)$$$$$$$$\mathrm{therefore}:$$$$f\mathrm{surjective}\iff \mathrm{\exists }x\in \mathbb{R}\mid f\left(x\right)=1$$$$\iff \mathrm{\exists }x\in \mathbb{R}\mid {\log }_{10}(\sqrt{3x^2-4x+\mathrm{k}+1}+10)=1$$$$\iff \mathrm{\exists }x\in \mathbb{R}\mid \sqrt{3x^2-4x+\mathrm{k}+1}+10={10}^1=10$$$$\iff \mathrm{\exists }x\in \mathbb{R}\mid \sqrt{3x^2-4x+\mathrm{k}+1}=0$$$$\mathrm{knowing}\mathrm{that}f\mathrm{must}\mathrm{be}\mathrm{well}\mathrm{defined},\mathrm{there}\mathrm{can}\mathrm{only}\mathrm{be}\mathrm{one}\mathrm{such}x$$$$\mathrm{otherwise},\mathrm{the}\mathrm{values}\mathrm{of}{3\mathrm{X}}^2-4\mathrm{X}+\mathrm{k}+1\mathrm{inbetween}\mathrm{would}\mathrm{be}\mathrm{negative}$$$$$$$$\mathrm{thus}:$$$$f\mathrm{surjective}\mathrm{and}\mathrm{well}\mathrm{defined}$$$$\iff \mathrm{\exists }!x\in \mathbb{R}\mid 3x^2-4x+\mathrm{k}+1=0$$$$\iff 4^2-4\times 3\times (\mathrm{k}+1)=0$$$$\iff 4-3(\mathrm{k}+1)=0$$$$\iff \mathrm{k}+1=\frac{4}{3}$$$$\iff \mathrm{k}=\frac{1}{3}{}_{◼}$$$$$$$$\mathrm{Answer}\left(\mathrm{A}\right)$$

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