-5) Assume that f : [a, b] → R is a continuously differentiable function on [a,...
Assume f : R" → R is twice continuously differentiable. Prove that the following are equivalent: (a) f(ex + (1-8)ì) < ef(x) + (1-8)/(x) for all x, x E Rn and 0 < θ < 1 (b) f(x)+ /f(x) . (x-x) -f(r) for all x,x E R" (c) f(x) > 0 for all x E R" Hint: Look at : RRdefine by gt) f(x + ty) where x, y E R. First show g is convex (as a function of...
5. [4 bonus points) Let F be a continuously differentiable non-decreasing function on R with F' = f. Show that 1,6)dF(a) = 5,5(x)}(a)dx 2 A A for every non-negative function g on R and a Borel set A.
5. Let f a, b R be a 4 times continuously differentiable function. For n even, consider < tn = b, a to < t< an uniform partition of [a, b] with b- a , i = 0,1,.. , n - 1 h t Let T denote the composite Trapezoidal rule associated with the above partition which approx imates eliminate the term containing h2 in the asymptotic expansion. Interprete the result which you obtain as an appropriate numerical quadrature rule...
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.
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'(x)|< A, for all x E R". Prove that the function F(x)= x - f(x) maps R" one-to-one and onto R". (Suggestion: Use the Contraction Mapping Principle Why not use the Inverse Function Theorem?)
differentiable function and there exists 0
4. (a) Assume a function h is differentiable at some point to. Is it true that h is continuous on some open-neighbourhood of xo? Provide either a proof or a counterexample. (b) Let f be twice differentiable on R and assume that f" is continuous. Show that for all x ER S(x) = S(0) + s°C)x + [ (x - 1))"(dt. (C) Deduce that for any twice continuously differentiable function f on R and any positive x > 0, x...
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...
Please prove by setting up the theorem below (Chain Rule)
v:RR is continuously differentiable. Define the Suppose that the function function g : R2R by 8(s, t)(s2t, s) for (s, t in R2. Find ag/as(s, t) and ag/at(s, t) Theorem 15.34 The Chain Rule Let O be an open subset of R and suppose that the mapping F:OR is continuously differentiable. Suppose also thatU is an open subset of Rm and that the functiong:u-R is continuously differentiable. Finally, suppose 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 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...