Refer to page 145 for problems on constrained optimization. Instructions: • Solve an optimization problem with constraints using the method of Lagrange multipliers. • • Interpret the significance of the Lagrange multipliers in the given context. Discuss the applications of this method in machine learning or operations research. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]

Give an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.

3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.

Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.

Solve this please

5.10. Discuss the continuity of the function f(z) = at the points 1, -1, i, and -i. 之一 21 3, |2 = 1

13.10. Let be the circle ||=2. Show that ཀ 4π V

2 Q /showthat Z +4 Z(Z-21) has aɣemovable discontinuity at z = 21.

13.4. Let f(z) =y-x-3ir² and y be given by the line segment z = 0 to z 1+i. Evaluate. L f(z)dz.