Prove-that-x-1-2-1-x-2-x-

Question Number 12173 by Mr Chheang Chantria last updated on 15/Apr/17

P r o v e t h a t \forall x \in [1, 2]

\Rightarrow 1 - x^{2} ⩽ x

Answered by ajfour last updated on 18/Apr/17

x ⩾ 1

\Rightarrow 1 - x ⩽ 0 ⩽ x^{2}

s o, 1 - x^{2} ⩽ x f o r x \in [1, \infty)

Commented by Mr Chheang Chantria last updated on 17/Apr/17

your solution is not enough

Answered by ajfour last updated on 17/Apr/17

l e t 1 - x^{2} ⩾ x

\Rightarrow x^{2} + x ⩽ 1

\Rightarrow x^{2} + x + \frac{1}{4} ⩽ \frac{5}{4}

o r {(x + \frac{1}{2})}^{2} ⩽ {(\frac{\sqrt{5}}{2})}^{2}

\Rightarrow - \frac{\sqrt{5}}{2} ⩽ x + \frac{1}{2} ⩽ \frac{\sqrt{5}}{2}

\Rightarrow - \frac{(\sqrt{5} + 1)}{2} ⩽ x ⩽ \frac{(\sqrt{5} - 1)}{2} < 1 < 2

s o i f x \in [1, 2] 1 - x^{2} < x .

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