Citadel Interview Question: Find the minimum of x^x

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Citadel Quantitative Researcher interview

Citadel Interview Question

Quantitative Researcher Interview

Find the minimum of x^x

Interview Answers

Anonymous

Jan 26, 2017

since taking a natural log preserves the structure of the function y = x^x, we get the following: 1) min[ ln(x^x) ] = min[x^x] 2) ln(x^x) = x ln(x) 3) d/dx[x ln(x)] = ln(x) + 1 = 0 We know it is minimized when derivative is zero if we observe the function of the derivative. Therefore, ln(x) = -1 x = 1/e

Anonymous

Apr 27, 2017

Denis's answer works for x>0, where ln() is defined. For x0 decays rapidly with the increase of k. When x = -(2k+1) (k in Z+), f(-(2k+1))=-1/(2k+1)^(2k+1)<0, decays in absolute value rapidly with k. Therefore, the smalest negative value is found for f(-1)=-1.

Anonymous

Oct 28, 2019

Small/pedantic edit of Denis's answer; in 1 the mins should be argmins. The argmin is 1/e, the min is (1/e)^(1/e)

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