Prove the inequality \(\displaystyle{e}^{{x}}\geq{x}^{{e}}\) for \(\displaystyle{x}{>}{0}\)

metalskaashw

metalskaashw

Answered question

2022-03-29

Prove the inequality exxe for x>0

Answer & Explanation

Eve Larson

Eve Larson

Beginner2022-03-30Added 9 answers

You are alsmot there.
Study the function f(x)=lnxx
Then f(x)=1x2(1lnx)
f(x)>0 for x<e and f(x)<0 for x>e
But f(e)=1e

Drake Huang

Drake Huang

Beginner2022-03-31Added 15 answers

You could define the function
f(x)=exxe
and show that f(x) is zero at x=1,e and go forth in terms of the minima of f(x). With this, however, you will also have to show that f(x) doesn't achieve a minimum at ±

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