Power Test

Algebra Level 3

True or False?

\quad There exists a real number x x such that x < x 3 < x 4 < x 2 . x<x^3<x^4<x^2.

True False

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Michael Fuller by Michael Fuller , 22, United Kingdom

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2 solutions

Akshat Sharda
Jul 26, 2016

Relevant wiki: Polynomial Inequalities - Problem Solving - Medium

x 3 − x > 0 ⇒ x ( x − 1 ) ( x + 1 ) > 0 ⇒ x ∈ ( − 1 , 0 ) ∪ ( 1 , ∞ ) x 4 − x 3 > 0 ⇒ x 3 ( x − 1 ) > 0 ⇒ x ∈ ( − ∞ , 0 ) ∪ ( 1 , ∞ ) x 4 − x 2 < 0 ⇒ x 2 ( x − 1 ) ( x + 1 ) < 0 ⇒ x ∈ ( − 1 , 0 ) ∪ ( 0 , 1 ) ∴ x ∈ ( − 1 , 0 ) x^3-x>0\Rightarrow x(x-1)(x+1)>0 \Rightarrow x \in (-1,0)\cup (1,\infty) \\ x^4-x^3>0\Rightarrow x^3(x-1)>0\Rightarrow x\in (-\infty,0)\cup (1,\infty) \\ x^4-x^2<0\Rightarrow x^2(x-1)(x+1)<0 \Rightarrow x\in (-1,0)\cup (0,1) \\ \therefore x\in (-1,0)

8 Helpful 0 Interesting 0 Brilliant 0 Confused
J D
Jul 26, 2016

Any value between zero and negative one, exclusive. -1/2 is an example.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Why not -1/2?

Peter van der Linden - 4 years, 10 months ago

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