Suppose that k e N and that f R"-R is homogeneous of order k: that is, that 'f(px)- kf(x) for all x є Rn and all є R. If f is differentiable on R", prove that af Экп af axi (Xi , . . . ,...
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...
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...
2. Suppose that A is an rn x n matrix and b є С". Prove that the linear system CSA, b) is consistent if and only if r(A) = r(Ab) 2. Suppose that A is an rn x n matrix and b є С". Prove that the linear system CSA, b) is consistent if and only if r(A) = r(Ab)
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0 for all x ∈ (0,∞). (a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈ N. (b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f '(k). (c) Let r > 1. By finding...
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...
1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove: f is differentiable at -2 f is differentiable at 1 2) Prove the product rule. Hint: Use f(x)g(x)− f(c)g(c) = f(x)g(x)−g(c))+f(x)− f(c))g(c). 3) Prove the quotient rule. Hint: You can do this directly, but it may be easier to find the derivative of 1/x and then use the chain rule and the product rule. 4) For n∈Z, prove that xn is differentiable and find the derivative, unless, of course, n...
Problem 3: Let xi be given n mutually orthogonal vectors in Rn, and 20 є Rn be also given. Find: (a) the distance di from Zo to Hi-{x E Rn : XTXǐ (b) the distance sk from ro to n1Hi, 1 <k< n (c) the distance mk from a'0 to ngk+1H,, 1-K n (d) calculate sk + mk 0) Problem 3: Let xi be given n mutually orthogonal vectors in Rn, and 20 є Rn be also given. Find: (a)...
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...
(8) Let E C R" and G C R" be open. Suppose that f E G and g G R', so that h = go f : E → R. Prove that if f is differentiable at a point x E E, and if g is differentiable at f (x) E G, then the partial derivatives Dihj(x) exist, for all and j - ...., and 7m に! (The subscripts hi. g. fk denote the coordinates of the functions h, g....
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentiable everywhere and calculate its total derivative (b) If f is differentiable everywhere and Df (x)A, for some A E Mp and all q.p x E Rp, show that there exists b E R, such that f(x) = Ax + b for all x E Rp 3. Let f: RP-R (a) If f(x)-Ax + b,...