Proving these two functions intersect at 'a'

[Saracen Rue]

Through experimental observations, I have found that the two functions $f\left(x\right)=x^{a\left(a-x^3\right)}-a$ and $g\left(x\right)=a^{x\left(x-a^3\right)}-x$ will always intersect at $a$ when $x>0$. Is there a way to mathematically prove this? For instance, simultaneously solving the functions and simplifying the answer down to $x=a$ (Note: the functions sometimes also intersect at other locations besides $[a, f(a)]$)

 

[PeroK]

Through experimental observations, I have found that the two functions $f\left(x\right)=x^{a\left(a-x^3\right)}-a$ and $g\left(x\right)=a^{x\left(x-a^3\right)}-x$ will always intersect at $a$ when $x>0$. Is there a way to mathematically prove this? For instance, simultaneously solving the functions and simplifying the answer down to $x=a$ (Note: the functions sometimes also intersect at other locations besides $[a, f(a)]$)

You mean prove that $f(a) = g(a)$?

 

[Saracen Rue]

You mean prove that $f(a) = g(a)$?

Ah yes, expressing it that way does seem a lot clearer.

 

[mfb]

Isn't that obvious if you just plug in a for x?

 

[Saracen Rue]

Right, so it was midnight when I posted this and I wasn't in the most clear mindset. Looking back on this now I can see that by proving $f(a)=g(a)$ you also prove that $f(x)$ intersects $g(x)$ at $x=a$. Sorry for this entire post.