(4) Let f R -R be a strictly conve:r C2 function and let 0 a) Write the Euler-Lagrange equation...
5. Euler and Lagrange developed the calculus of variations. (a) Explain in a sentence what the calculus of variations is about. (b) Find the Euler-Lagrange equation (assuming y(0) 0 and y(1) 0 for minimizing the integral: (ul(t)(t)) dt. + (y(t))2 Recall that for the Lagrangian density function L(t, y, u), the Euler-Lagrange equation is as a function of t where u(t) = t). (c) How is the Euler-Lagrange equation in general related to the directional derivative of vector calculus? Why...
5. Let f(x) = ax2 +bx+c, where a > 0. Prove that the secant method for minimization will terminate in exactly one iteration for any initial points Xo, X1, provided that x1 + xo: 6. Consider the sequence {x(k)} given by i. Write down the value of the limit of {x(k)}. ii. Find the order of convergence of {x(k)}. 7. Consider the function f(x) = x4 – 14x3 + 60x2 – 70x in the interval (0, 2). Use the bisection...
7. [8 POINTS] Let f: R → R be a strictly increasing function. Prove by way of contradiction that there cannot be more than one place where f crosses the x-axis.
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(r, 0) f(a) w(0, t) g(t) w(1, t) h(t) for a function w: [0,1 x [0, 0)- R such that w, wn, and wa exist and are continuous. Show that the solution to this problem is unique, that is, if w1 and w2 [0, 1] x [0, 00)- R both satisfy these conditions, then w1 = w2....
Let a continuously differentiable function f: Rn → R and a point x E Rn be given. For d E Rn we define Prove the following statements: (i) If f is convex and gd has a local minimum at t-0 for every d E R", then x is a minimiser of f. (ii) In general, the statement in (i) does not hold without assuming f to be convex. Hint: For) consider the function f: R2-»R given by Let a continuously...
(a) Show that, if y satisfies the Euler-Lagrange equation associated with the integral 2. qy2) dx, I = (6) where p() and q(x) are known functions, then I has the value 12 (b) Show that, if y satisfies the Euler-Lagrange equation associated with (6) and if z(x) is an arbitrary differentiable function for which z(x)z(r2) = 0, (7) 1 then (yyz)da= 0. + Hence show that by replacing y in (6) by the function (y + z), where the condition...
PROVE: 4. If f : R → R is a strictly increasing function, f(0) = 0, a > 0 and b > 0, then
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
Let f : [0,∞) → R be the function defined by f ( x ) = 2 ⌊ x ⌋ − x? where x? = x − ⌊x⌋ is the decimal part of x. Prove that f is injective. Let f: 0,00) + R be the function defined by f(3) = 212) where ã = x — [x] is the decimal part of x. Prove that f is injective.
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction of V f(a), and the maximum rate of increase is Vf(a)l (a) Prove that for each unit vector u e R" (b) Prove that if ▽/(a)メ0, and u = ▽f(a)/IV/(a) 11, then Let U be an open subset of R". Let...