Matrix chain multiplication is a well known example that demonstrates utility of dynamic programming.

Matrix chain multiplication is an optimization problem which is that can be solved using dynamic programming.

They showed how the matrix chain multiplication problem can be transformed (or reduced) into the problem of partitioning a convex polygon into non-intersecting triangles.

MPF or mariginalize a product function is a general computational problem which as special case includes many classical problems such as computation of discrete Hadamard transform, maximum likelihood decoding of a linear code over a memory-less channel, and matrix chain multiplication.