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In optimal control, the situation is more complicated because of the possibility of a singular solution.The generalized Legendre–Clebsch condition, [1] also known as convexity, [2] is a sufficient condition for local optimality such that when the linear sensitivity of the Hamiltonian to changes in u is zero, i.e.,
The films are made primarily in the Tamil language. Chintamani (1937) and Haridas (1944) are landmark films from the early stages of Tamil cinema. Chintamani was the first major box office success, while Haridas is noted for its significant impact and long theatrical run, reflecting industry's development before its formal establishment post ...
If the domain is just the real line, then () is just the second derivative ″ (), so the condition becomes ″ (). If m = 0 {\displaystyle m=0} then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that f ″ ( x ) ≥ 0 {\displaystyle f''(x)\geq 0} ), which implies the function is convex, and ...
The Tamil-language film industry has produced and released over 10000 films since the release of its first sound film, kalidas, in 1931.This is a master list of listings of films released from 1931 in the industry by decade of release.
The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see 'Second derivative test'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.
Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upper-leftmost) minors (determinants of sub-matrices) of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization—the ...
At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f .)
The following is the list of highest-grossing Tamil cinema films released in 2024. The rank of the films in the following table depends on the estimate of worldwide collections as reported by organizations classified as green by Wikipedia. [a] There is no official tracking of domestic box office figures within India.