Yesterday, I accidentally discovered a nasty bug in a fairly simple example:
I am sure the correct answers are sqrt(a^2+b^2) and -sqrt(a^2+b^2) for any real values a and b . It is easy to prove in many ways. The simplest method does not require any calculations and can be done in the mind. We will consider Expr as the scalar product (or the dot product) of two vectors <a, b> and <sin(x), cos(x)>, one of which is a unit vector. Then it is obvious that the maximum of this scalar product is reached if the vectors are codirectional and equals to the length of the first vector, that is, sqrt(a^2+b^2).
Bugs in these commands were noted by users and earlier (see search by keywords bug, maximize, minimize) but unfortunately are still not fixed.