Unimodality of continuous univariate distributions What are the different methods to prove that a continuous univariate distribution is unimodal?
Some of them are available at Wikipedia but not much detail is given. Solution or suggestion of any continuous univariate distribution is welcome.    
 A: Suppose the distribution is defined by a function $f(x)$.  The obvious method is to use calculus to find all local maxima of the function.  It has a local maxima at $x = x^*$ if $f'(x^*) = 0$ and $f''(x^*) < 0$, where $f'$ is the first derivative and $f''$ the second dervivative.  Having identified all values of $x$ corresponding to local maxima (often there will only be one), find the associated maximum values of $f$ and compare them.  If one of the local maxima is greater than any others, then the distribution is unimodal (although as the Wikipedia article notes the terminology is not always consistently used and a distribution with two local maxima, one greater than the other, is sometimes described as bimodal).
Note however that the above method will not work in a case where the function, though continuous, is not differentiable at all values of $x$.  An example is the Laplace distribution, which is not differentiable at its mode. In such a case it may be best just to calculate and plot the value of the function at regularly spaced values of $x$ to determine its approximate shape. 
