Determine-whether-the-following-is-true-for-all-value-of-x-0-x-1-2-x-2-x-1-4-3-

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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