Without direct evaluation, determine which of these following expressions is larger:
(φ)√2 and (√2)φ
Assume it is known that 1<√2<φ<2, but not their exact values.
Call the expressions:
(φ)^√2 (I)
(√2)^φ (II)
Multiply I and II by (√2) and square:
((√2)*((φ)^√2))^2 = 2φ^(2√2)
((√2)*((√2)^φ))^2 = 2^(φ+1)
Divide by 2:
φ^(2√2)
2^(φ)
Let φ = √2+k, with k small.
Let x=(√2+k)^(2√2)-2^(√2+k)
If k=0, then x=√2^(2√2)-2^(√2)=0
Any k larger than this will clearly yield a positive result, e.g. if k=(2-√2) then
(√2+2-√2)^(2√2)-2^(√2+2-√2) = 2^(2 √2) - 2^2, positive.
So expression I is larger than expression II
Edited on June 1, 2024, 11:09 am
|
|
Posted by broll
on 2024-06-01 11:08:15 |
Possible Solution