Determine what value of a allows for largest probability

[Saracen Rue]

Homework Statement

There are two possible solutions for 'a' for which $f(x)=|x^x-x^a|$ is a probability density function. Determine the value for a which produces a PDF with the largest probability of random variable 'x' falling within two standard deviations either side of the mean.

Homework Equations

Just the rules to do with determining if function is a PDF using integrals as well as finding the mean, variance and standard deviation.

The Attempt at a Solution

I'm honestly at a loss here. There's no way to find the definite integral of this function meaning we need to integrate over a set interval. I calculated the x-intercepts to be 1 and a. I tried getting my calculator to solve (for a) $∫_0^1f(x)dx=1$ and $∫_0^af(x)dx=1$, both individually and simultaneously, but my calculator kept giving me an error. I'm really not sure what to do here - is there a way to do this manually that I don't knoe about? Any help is greatly appreciated :)

 

[mfb]

As first step I would try to numerically estimate where these values of a are.
I found one so far, I'm not so sure where to look for the other one.

 

[Saracen Rue]

As first step I would try to numerically estimate where these values of a are.
I found one so far, I'm not so sure where to look for the other one.

Thank you for your help - I had considered just brut forcing it by au substituting different values of a in but I had assumed there was an easier way to do it. I used an online graphing calculator to hell me visualise what was going on and to know where to look for the values of a. If you wouldn't mind could you please look over my working out for this? Here's the link: https://www.desmos.com/calculator/68gbrixyqf
Note, I defined to different functions for each value of a while I was working this out to keep better track of things. I very much appreciate you spending your time to help :)

 

[mfb]

Maybe the whole question asks for a numerical analysis. Just by eye I'm quite sure I know which one has a larger integral within 2 standard deviations, but that is not a very mathematical approach.