I would like to get some explanation or intuition behind the process of finding the maximum likelihood of an exponential distribution.

Given the following likelihood function:

$Lleft( theta right) = theta ^{4}e^{-8.65theta }$

I would like to find $theta$ for which the likelihood is maximised. I know I can achieve this by taking the derivative and solving the equation:

$dfrac{dLleft( theta right) }{dtheta }=0$

Having calculated the derivative I obtained:

$dfrac{dLleft( theta right) }{dtheta } = theta ^{3}left( 4-8.65theta right) cdot e^{-8.65theta }$

Solving it for $0$ yields ~ $0.46$.

When one takes a look at the plot of this function, one can see that this function doesn’t have the maximum in that place. Since it is $e^(-x)$, the max value goes to the inf.

What am I actually determining, if I say that likelihood is maximised at 0.46?