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Proposition 19 (The First-Derivative Test for a) changes its sign from If at a stationary positive to negative, point x0 the first b) changes its sign from derivative of a negative to positive, function f c) does not change its sign, Local Extremum) then the value of the function at x0 , f (x0 ), will be a) a local maximum; b) a local minimum; c) neither a local maximum nor a local minimum. Proposition 20 (Second-Derivative Test for Local Extremum) A stationary point x0 of f (x) will be a local maximum if f (x0 ) < 0 and a local minimum if f (x0 ) > 0.

M. Solve these equations for x1 , . . , xn and λ1 , . . , λm . In the end we will get a set of stationary points of the Lagrangian. If x∗ = (x∗1 , . . , x∗n ) is a solution of (4), it should be a stationary point of L. It is important that rank(J ) = m, and the functions are continuously differentiable. Example 66 This example shows that the Lagrange-multiplier method does not always provide the solution to an optimization problem. Consider the problem: min x2 + y 2 subject to (x − 1)3 − y 2 = 0.

If A1 is the matrix representation of T in the basis B, A2 is the matrix representation of T in the basis C and D is the change-of-basis matrix relative to C, B then A2 = D−1 A1 D. Further Reading: • Bellman, R. Introduction to Matrix Analysis. B. A. Beauregard. Linear Algebra. R. The Theory of Matrices. • Lang, S. Linear Algebra. 1 The Concept of Limit Definition 29 The function f (x) has a limit A (or tends to A as a limit) as x approaches a if for each given number ε > 0, no matter how small, there exists a positive number δ (that depends on ε) such that |f (x) − A| < ε whenever 0 < |x − a| < δ.

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