When finding the maximum likelihood of a function, we take the derivative and set it to 0.
If the density f(x) is smooth, the log density should be smooth also. The linear combination of smooth log functions is also smooth. Then the loglikelihood is smooth. If there is only one solution to the parameter, call it $\hat{\theta}$, then we can say it's either a maxima or a minima, since if it was an inflection point that would imply the likelihood goes to infinity.
Then we can just plug in any point $\theta'$ and test if the likelihood is higher/lower than $\hat{\theta}$, if $L(\theta')$ is lower then $L(\hat{\theta})$ then $\hat{\theta}$ is a maxima and visa versa. So we don't need to check the second derivative.
Is this right?