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- Maximize the function fx,y=7x+5y in the region determined by the constraints of Problem 34.Maximize f(x) = 4x 1.8x² + 1.2x³- 0.3x4 a) Using Golden-Section Search (x₁ = −2,xu = 4, εs = 1%) b) Using Newton's Method (xo = 3, &s=1%)Analyze the stability of the critical point of the following system, proposing as a Lyapunov function V(x,y)=ax2m+by2n. a) x′=−2xy, y′=x2−y3 b) x′=xy2−(x3/2), y′=−(y3/2)+(yx2/5) c) Show that x′=−2x+xy3, y′=−x2y2−y3 is asymptotically stable. Pleas be as clear as possible and legible. Show and explain all the steps in detail. Thank you very much.
- Find all critical points of the the functionF(x,y)=ln(x2 +y2 +2y+2) in the domain D = {(x,y) ; x2 + y2 < 16}, and decide if each is a local minimum, local maximum or saddle point.Analyze the stability of the critical point of the following system, proposing as a Lyapunov function V(x,y)=ax2m+by2nThe function f (x, y) = x³ + y³ + 3x² - 3y² - 8 has local minimum at the critical point (0, 2). True or False
- Consider the following quadratic function: f(θ) = 1/2∑ni=1 wi(θ−xi)2, where x1,...,xn are points in 1D space taking on real values and w1,..,wn are positive real numbers denoting the importance of each of these points. What value of θ minimizes f(θ)?Find the quadratic function f (x) = ax2 + bx + c that goes through (0, 1) and has a local minimum at (1,-1).Find the critical points for the function f(x,y)=x3+y3−9x2−27y+6 and classify each as a local maximum, local minimum, saddle point, or none of these.
- For the function f(x,y)=x^2+2y^2−(x^2)y, does the critical point (−2,1) correspond to a local maximum, a local minimum, or a saddle point?Find two positive numbers x and y with xy=81 such that x+4y is minimized.Use Lagrange multipliers to find the critical points and the relative extrema. f(x,y,z) = x² + y² + z² subject to the constraint 0 = x² - y² + 1