Solving the exponential inequality (sqrt2−1)^x+(sqrt2+1)^x<=2 without using logarithms or calculus

hotonglamoz

hotonglamoz

Answered question

2022-09-27

Solving the exponential inequality ( 2 1 ) x + ( 2 + 1 ) x 2 without using logarithms or calculuы

Answer & Explanation

elilsonoulp2l

elilsonoulp2l

Beginner2022-09-28Added 9 answers

You have
2 1 = 1 1 + 2
So, basically, you look for a > 0 with
1 a + a 2
Fact is, that because of, for example, AM-GM you have
1 a + a 2
with equality if and only if a=1.
So, the only solution to your inequality is
a = ( 1 + 2 ) x = 1 x = 0

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