Assume f : R" → R is twice continuously differentiable. Prove that the following are equivalent: ...
linear optimization
Assume that f : D → R is twice continuously differentiable for all x D, where the domain D off is an open, convex subset of Rn. Sh ▽2f(x), is symmetric positive-semi-definite for all x E D if and only if f is a convex function on D Moreover, if its Hessian matrix. ▽2 (x), is symmetric positive-definite for all x E D, then f is a strictly convex function on D Show that the converse of this...
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...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
3. (25 pts) Suppose f(x) is twice continuously differentiable for all r, and f"(x) > 0 for all , and f(x) has a root at p satisfying f'(p) < 0. Let p, be Newton's method's sequence of approximations for initial guess po < p. Prove pi > po and pı < p Remember, Newton's method is Pn+1 = pn - f(pn)/f'(P/) and 1 f"(En P+1 P2 f(pP-p)2. between pn and p for some
3. (25 pts) Suppose f(x) is twice...
Now assume that f(0) = 0 and f'(0) = 0. Prove that if f is twice differentiable and If"(x) < 1 for all x E R then 22 Vx > 0, f(x) < 2
-5) Assume that f : [a, b] → R is a continuously differentiable function on [a, b] with f(a) = f(6) = 0 and x dx = 1. Prove: (2) f'(x) dx = -1/2, and [cm)? ds. [ f(a)dx > 1/4
4. Prove the following statement: Consider the ODE x = f(x) with x : J C R → Rn and f : Rn → Rn. If a continuously differentiable real-valued function V = V(x) exists such that (a) V is defined on Bs(0) {x E Rn : Irl < δ} (b) V(x) 0 for x E Bs(0) 1 fo) (c) V 0) 1 (o then the origin is unstable. (x) >0 for rE Bs
4. Prove the following statement: Consider...
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)0 or all x, y E 5) S= (xe(-1,4] Sin(x) > 0). Let of the following is true? I. inf (S).< 0 2. sup (S) does not exist Which . sup (S) π ,' inf (S) = π/2
1 Let f: R R be...
4. Let F be a continuously differentiable function, and let s be a fixed point of F (a) Prove if F,(s)| < 1, then there exists α > 0 such that fixed point iterations will o E [s - a, s+a]. converge tO s whenever x (b) Prove if IF'(s)| > 1, then given fixed point iterations xn satisfying rnメs for all n, xn will not converge to s.
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that:
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...