Determine whether the following is true for all value of x 0≤(((x+1)^2 )/(x^2 +x+1))≤(4/3)


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Question Number 50890 by peter frank last updated on 21/Dec/18

$D e t e r m i n e w h e t h e r$ $t h e f o l l o w i n g i s t r u e f o r a l l$ $v a l u e o f x$ $0 ⩽ \frac{{(x + 1)}^{2}}{x^{2} + x + 1} ⩽ \frac{4}{3}$
	Answered by MJS last updated on 21/Dec/18

	$\forall x \in R : {(x + 1)}^{2} ⩾ 0 \land (x^{2} + x + 1) ⩾ 0 \Rightarrow 0 ⩽ \frac{{(x + 1)}^{2}}{x^{2} + x + 1}$ $3 {(x + 1)}^{2} ⩽ 4 (x^{2} + x + 1)$ $3 x^{2} + 6 x + 3 ⩽ 4 x^{2} + 4 x + 4$ $0 ⩽ x^{2} - 2 x + 1$ $0 ⩽ {(x - 1)}^{2} \forall x \in R \Rightarrow$ $\Rightarrow 0 ⩽ \frac{{(x + 1)}^{2}}{x^{2} + x + 1} ⩽ \frac{4}{3} \forall x \in R$
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