T(ii) Show that the quadratic function f(x) = x" Ax+2b7x+c with A = A" ER²x",bER",cER is strongly convex if and only if A > 0, and in that casethe strong convexity parameter is 21min(A).%3|

Answered: (ii) Show that the quadratic function…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Day: Date: //20 A constrained extremisation Problem is expressed as longsonge-multiplier problem with objective functional. I [X³ X] = √ [√x² ² + y² + 3 2 _k (t) (x²+ y² +3²_R²) ]dt where x = dxc etc. and intereral has appropriate fixed limitat with fixed values of Xoy and 7 at the endpoints. a). Explain what geometric problem is solved fiere and Is there a local or global constraint. for Variable x b)- show that ewler-longronge eq. takes the form, d at x √ ( 7²7 5²,50) = -2/x x з Also find Euler -Longronze ey's for I and 3 land write them in vector notation for vector X = (x₂y₂3). Give Jemeterical interpretation what they tell you. C). show that Beltrami-identity does not help you to solve the equation- d). Let u be an arbitrary increasing for of t. show that ƒ [√xx ²)² + (Y²) ²³+ (3³) ³ ] du = √ [] x² + y + z ² ] du where. x'=dx, Explain what this signifies geometrically and why this means that one can assumes without loss of generalify that | 1 = R along solution of the |…
12. Determine the local minima, local maxima, and saddle points of f(x, y) = x² – 4ry + 2yª + 1.
4. The quadratic function f(x) has a turning point at (-3,6). The quadratic y=f(x)+3 would have a turning point of (1) (-2,9) (3) (-3, 7) (2) (1, 7) (4) (–1, 9)
4. Find a quadratic function f(x) = ax² + bx + c which has a vertex at (1, 1) and passes through (3, –3). (Hint: first find the standard equation of this parabola and then expand this standard equation.)
Suppose f(): R" - R is a function. Consider the optimization problem min f(x) over x ER". (P) #2) Suppose f(-) € C¹ and x solves (P) (i.e. Ỹ € argmin(P)). (a) Explicitly, what does X € argmin(P) mean? (b) What is the first order necessary condition for x = argmin(P)? (c) Prove the first order necessary condition.
9. Let f(x) = 3x°(x+ 3)(x – 1) . a) Find all x-intercepts for this function. b) A polynomial function with the same degree as this function might have as few as x-intercepts(s) but cannot have more than x-intercept(s). c) A polynomial function with the same degree as this function might have as few as "peak(s)" and/or "valley(s)" but cannot have more than "valley(s)". d) When |x| is large, the graph of this function resembles the graph of y = ax", where a = "peak(s)" and/or and n =
1. Find the local extremum of the function (a) f(x,y) = 2x² – xy + y² – x – 5y + 8 | (b) f(x,y) = xy² – 6x² – 3y² %3D - əz and Əx az 2. Find if the function is implicitly defined as a function of x and y. ду уех2 — Зуеу2 %3D — 4ze*у + 1 - 3. Given f(x, y) = xy + ey Verify that fry = fyx y2 + 1 4. Given f(x, y) = /9 – x² – y2. Use the total differential to approximate %3D '9 – 2.022 – 0.992 – 9 – 22 – 12 -
USE THE ABOVE EQUATION TO SOLVE THE PROBLEM
4. Letf &g be the functions defined by below by f(x)=√x-e² + (x+1)(x² - 1) & g(x) = (x²-2 6x-11 x < 3 x ≥ 3
5. Sketch the quadratic f(x) = (x-4)² + 1 by moving the 5 main points from the parent function, y = x² in the appropriate manner. 10-
F(t)=t-5 and g(t)=t-3, find

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(ii) Show that the quadratic function f(x)=x" Ax+2b"x+c with A = AT E R²x",bER",c ER is strongly convex if and only if A > 0, and in that case the strong convexity parameter is 22min (A).