[![alt text][1]][1]  
<sub>(source: [uci.edu](http://vmm.math.uci.edu/graph.jpg))</sub>  


  [1]: https://i.sstatic.net/uPioZ.jpg
No, consider a function $f$ with a graph something like the above for example. (Pardon the rough sketch:-)   When you mollify it the minimum at the origin will rise and become greater that the value of the mollified function away from the origin.