How to prove monotonicity of this function? Let a>0. How to prove that the function: P

Laney Spears

Laney Spears

Answered question

2022-01-22

How to prove monotonicity of this function?
Let a>0. How to prove that the function:
f(x)=ax1log(ax)logxx1,
is monotonic (depending on a<1 or a>1). I know that we can calculate the derivative and determine its sign, but this needs much of calculation. I'm wondering if we can decide the monotonicity using a simple trick.

Answer & Explanation

Ronald Alvarez

Ronald Alvarez

Beginner2022-01-23Added 11 answers

Consider
g:RR   g(t)=lnet1t
(where the value at 0 is defined by continuity). Then from
lnf(x)=g(lnx+lna)g(lnx)
we know that 1lna is monotone for all a1 if g is convex. But g's second derivative
g(t)=cosht1t22t2(cosht1)
is nonnegative.
Admittedly, since
g(t)=t+n=1Bnntnn!
is itself an important elementary function, using the second derivative to reduce the result to inequalities of other elementary functions seems unsatisfactory. Maybe there's a more intrinsic route?

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