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

Prove that ∀x∈[1,2] ⇒ 1−x^2 ≤ x

Provethat∀x∈[1,2]⇒1−x2⩽x

Answered by ajfour last updated on 18/Apr/17

x≥1 ⇒ 1−x ≤ 0 ≤ x^2 so, 1−x^2 ≤ x for x∈ [1,∞)

x⩾1⇒1−x⩽0⩽x2so,1−x2⩽xforx∈[1,∞)

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

your solution is not enough

yoursolutionisnotenough

Answered by ajfour last updated on 17/Apr/17

let 1−x^2 ≥ x ⇒ x^2 +x ≤ 1 ⇒ x^2 +x+(1/4) ≤ (5/4) or (x+(1/2))^2 ≤( ((√5)/2))^2 ⇒ −((√5)/2) ≤ x+(1/2) ≤ ((√5)/2) ⇒ −((((√5)+1))/2) ≤ x ≤ ((((√5)−1))/2) < 1 < 2 so if x∈ [1,2] 1−x^2 < x .

let1−x2⩾x⇒x2+x⩽1⇒x2+x+14⩽54or(x+12)2⩽(52)2⇒−52⩽x+12⩽52⇒−(5+1)2⩽x⩽(5−1)2<1<2soifx∈[1,2]1−x2<x.

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