How to prove monotonicity of this function?Let a&gt;0. How to prove that the function:f(x)=ax−1log⁡(ax)⋅log⁡xx−1,is monotonic (depending on a&lt;1 or a&gt;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.

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How to prove monotonicity of this function?Let a&amp;gt;0. How to prove that the function:f(x)=ax−1log⁡(ax)⋅log⁡xx−1,is monotonic (depending on a&amp;lt;1 or a&amp;gt;1). I know that we can calculate the derivative and determine its sign, but this needs much of calculation. I&#039;m wondering if we can decide the monotonicity using a simple trick.

Ronald Alvarez · Accepted Answer

Consider g:R→R   g(t)=ln⁡et−1t (where the value at 0 is defined by continuity). Then from ln⁡f(x)=g(ln⁡x+ln⁡a)−g(ln⁡x) we know that 1ln⁡a is monotone for all a≠1 if g is convex. But g&#039;s second derivative g′′(t)=cosht−1−t22t2(cosht−1) is nonnegative. Admittedly, since g(t)=t+∑n=1∞Bnntnn! is itself an important elementary function, using the second derivative to reduce the result to inequalities of other elementary functions seems unsatisfactory. Maybe there&#039;s a more intrinsic route?

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

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