Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

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.

Terms of Service

Privacy Policy

Contact: info@tinkutara.com