Let $ X_1, ... X_n$ i.i.d with pdf
$$f(x;\theta)=\frac{x+1}{\theta(\theta+1)}\exp(-x/\theta), x>0, \theta >0$$
It is asked to find the MLE estimator for $\theta.$
The likelihood function is given by
$$L(\theta;x)=[\theta(1-\theta)]^{-n}\exp\left(-\frac{\sum_i x_i}{\theta}\right)\prod_i (x_i+1)I_{(0,\infty)}(x_i)$$
Then, the derivative of log-likelihood will be
$$\frac{dlogL(\theta;x)}{d\theta}=\frac{-(2\theta+1)n}{\theta(1+\theta)} + \frac{\sum X_i}{\theta^2}$$
I've obtained my candidate to MLE. But, doing the second derivative of the log-likelihood, I could not conclude that it is negative so that the candidate is, indeed, the point of maximum. What should I do, in this case?
Thanks in advance!