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Multivariable Optimization

How we use multivariable calculus for local maxima, minima, and saddle points for a differentiable C^2 function z=f(x,y). Finding critical points and testing them. (Multivariable Calculus Unit 3 Lecture 15) Recall: Clairaut's theorem, which asserts that for 𝐶2 functions, the mixed second-order partial derivatives can be computed in any order, meaning 𝑓𝑥𝑦=𝑓𝑦𝑥. If the gradient of a function at a point is the zero vector, then the directional derivative of the function in any direction at that point is zero. This implies that if we are at a point (𝑎,𝑏) in the domain and the gradient is zero, the function appears flat when observed from that point. Definition of Critical Points: We define critical points of a function as points where the gradient is zero. These points are candidates for local extrema or saddle points. Through a graphical representation, such as the MATLAB peaks surface, we can visually infer the locations of local maxima, minima, and saddle points. We observe that at local extrema, the tangent planes to the function's graph are flat. In other words, the orthogonal (or "normal") sense of direction is straight up and down. We note that the normal vector to these tangent planes can be represented as −∇𝑓=(−𝑓𝑥,−𝑓𝑦,1). At extrema, this gradient vector is zero. We focus on the determinant of the Hessian matrix, given by: 𝐷=𝑓𝑥𝑥𝑓𝑦𝑦−(𝑓𝑥𝑦)^2 If 𝐷 and 𝑓𝑥𝑥 are positive at a critical point, it's a local minimum. If 𝐷 is positive and 𝑓𝑥𝑥 is negative, it's a local maximum. If 𝐷 is negative , the point is a saddle point. If 𝐷=0, the test is inconclusive. Example: Function 𝑓(𝑥,𝑦)=2𝑥^2−4𝑥𝑦+𝑦^4+2 We first find the critical points by setting the gradient to zero: ∇𝑓(𝑥,𝑦) = ⟨4𝑥−4𝑦 , −4𝑥+4𝑦^3⟩ = 4⟨𝑥−𝑦,𝑦3−𝑥⟩ This leads to the critical points (0,0), (1,1), and (−1,−1). We compute the determinant of the Hessian matrix and apply the second derivative test at each critical point. We conclude that (0,0) is a saddle point, while (1,1) and (−1,−1) are local minima. At this point, we have established a method for identifying and classifying local extrema of scalar-valued 𝐶^2 functions of two variables. We note that this method identifies local, not absolute, extrema. The next lesson will bring in absolute extrema, further enriching our understanding of optimization in multivariable calculus. #calculus #multivariablecalculus #mathematics #iitjammathematics #optimization #partialderivatives #calculus3