An optimal parenthesization of a matrix-chain product is one that minimizes the number of scalar multiplications required to compute the product. In general, finding an optimal parenthesization for a given sequence of matrix dimensions can be a complex problem, and there are several algorithms that can be used to solve it.
One possible optimal parenthesization of the matrix-chain product with the given sequence of dimensions is as follows:
((((5 x 10) x 3) x 12) x 5) x 50) x 6
This parenthesization involves a total of 45 scalar multiplications, which is the minimum number of multiplications required to compute the product. To verify that this is indeed an optimal parenthesization, it would be necessary to compare it to other possible parenthesizations and calculate the number of scalar multiplications required in each case. However, without additional information or specific algorithms, it is not possible to determine whether this parenthesization is optimal with certainty.