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If-xy-ax-2-2y-2-is-always-true-for-any-1-x-2-2-y-3-Then-find-the-range-of-a-

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Question Number 184618 by CrispyXYZ last updated on 09/Jan/23
If xy≤ax^2 +2y^2 is always true for any 1≤x≤2, 2≤y≤3 Then find the range of a.
Ifxy⩽ax2+2y2isalwaystrueforany1⩽x⩽2,2⩽y⩽3Thenfindtherangeofa.
Answered by mr W last updated on 09/Jan/23
let t=(y/x) 2t^2 −t+a≥0 1≤t≤3 case (A) Δ=1−8a≤0 ⇒a≥(1/8) or Δ=1−8a≥0 ⇒a≤(1/8) and case (C) case (B) ((1−(√(1−8a)))/4)≥3 or ((1+(√(1−8a)))/4)≤1 −(√(1−8a))≥^(never) 11 or a≥−1 summary: a≥(1/8) or −1≤a≤(1/8), i.e. a≥−1
lett=yx2t2−t+a⩾01⩽t⩽3case(A)Δ=1−8a⩽0⇒a⩾18orΔ=1−8a⩾0⇒a⩽18andcase(C)case(B)1−1−8a4⩾3or1+1−8a4⩽1−1−8a⩾never11ora⩾−1summary:a⩾18or−1⩽a⩽18,i.e.a⩾−1
Commented by SEKRET last updated on 09/Jan/23
−15≤a≤(1/8)
−15⩽a⩽18
Commented by mr W last updated on 10/Jan/23
Commented by mr W last updated on 09/Jan/23
if −15≤a≤(1/8) is right, then a=1 is wrong. but 2t^2 −t+1 is always >0 for 1≤t≤3. that means a=1 is right.
if−15⩽a⩽18isright,thena=1iswrong.but2t2−t+1isalways>0for1⩽t⩽3.thatmeansa=1isright.
Commented by mr W last updated on 09/Jan/23
if −15≤a≤(1/8) is right, then a=−15 is right, but 2t^2 −t−15 is <0 for 1≤t≤3, that means a=−15 is wrong.
if−15⩽a⩽18isright,thena=−15isright,but2t2−t−15is<0for1⩽t⩽3,thatmeansa=−15iswrong.

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