4.3, #11. Suppose that f is differentiable on (a, b), and if f' takes on two...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
3. In this problem we shall investigate the intermediate value theorem for derivatives. (a) Differentiate the function f(c)= sin ), 2 0 = 0,1=0 Show that f'(0) exists but that f' is not continuous at 0. Roughly sketch f' to see that nevertheless, f' doesn't seem to "skip any val- ues". Now let f be any function differentiable on (a, b) and let 21,22 € (a, b). Suppose f'(21) < 0 and f'(22) > 0. (b) By the Extreme Value...
9. Suppose that f : [0,-) + R is differentiable and that the derivative f' : [0,00) + R is also differentiable, with f(0) = f'(0) = 0. Suppose also that [f"(x) < 1 for all € [0, 0). a) Show how the Mean Value Theorem can be used to prove that f(x) <r? for all x € (0,00). b) Show how the Cauchy Generalized MVT can be used to prove a stronger statement: |f(7) < 2 for all 2...
real analysis 1,2,3,4,8please 5.1.5a Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = In(x), (1,91 Yes, it does not matter if is continuous or differentiable, every function satisfies the Mean Value Theorem. Yes, f is continuous on [1, 9] and differentiable on (1,9). No, f is not continuous on 1, 9). No, f is continuous on [1, 9] but not differentiable on (1,9). There is not enough information to verify if this function satisfies the Mean...
1. The function f has derivative f' where f' is increasing and twice differentiable. Selected values of f' are given in the table above. It is known that f(0) = 3. (a) For f'(x), the conditions of the Mean Value Theorem are met on the closed interval (0,3). The conclusion of the Mean Value Theorem over the interval (0,3) for f'(x) is satisfied at c = 1. Find f"(c). (b) Use a right Riemann sum with the three subintervals indicated...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
Under is for reference (Mean Value Theorem): Suppose that f: R6 + R is a function with the following two properties: flo) = 0, and at at any point Te R6 and any increment h, || DFOD | E || ||. Show that f(B1)) (-1,1). Comment. You should use the Mean Value Theorem at some point in this problem. An interpretation with more jargon is that if the operator norm of Df is at most 1 at all points, then...
a. Determine whether the Mean Value Theorem applies to the function f(x) = x + on the interval [3,6]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. a. Choose the correct answer below. O O A. No, because the function is not continuous on the interval [3,6], and is not differentiable on the interval (3,6). B. No, because the function is differentiable on the interval (3,6), but is not continuous...
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...