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2. Rolle's theorem states that if F : [a, b] → R is a continuous function,...
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,...
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) =...
Problem 24. Suppose the function f and its derivative f' are continuous on [a,bl. Let s be the are length of the curve f from the point (a, f(a)) to (b,f(b)). 1. Let a =x0 < 시<x2 < <x,' = b be a partition ofla,bl. 2. Show that s = 1 + Lr'(x) dx by using the Mean Value Theorem for differentiation
a) Verify the Rolle's theorem for the function f(x) = -1 x +x-6 over the interval (-3, 2] 3-X b) Find the absolute maximum and minimum values of function f(x)= (1+x?)Ě over the interval [-1,1] c) Find the following for the function f(x) = 2x – 3x – 12x +8 i) Intervals where f(x) is increasing and decreasing. ii) Local minimum and local maximum of f(x) iii) Intervals where f(x) is concave up and concave down. iv) Inflection point(s). v)...
how do i solve this with Mean Theorem Value? 4. 0.5/1 poilnts 1 Previous Anawars LarCal: 11 3.2 020 Determine whether Rolle's Theorem can be applied to f on the closed interval bl. (select all that apply.) Yes O No, because fis not continuous on the closed Interval [a, bl ND, hecause fis rnot differentiable in the open interval (a, b). No, because ) If Ralle's Theorem can be applied, find all values of cin the open interval (a, b)...
Logic (a) Let f : [a, b] → R be a continuous function. Prover that there exists ce [a, b] such that con la silany - be a criminatoria per le sue in elan 5(e) = So gladde (b) Define F:R+R by F(x) = [** V1+e=i&t. Prove by citing the appropriate theorem(s) that F is differentiable on R, and calculate F'(c). Be sure to justify your reasoning at every stage.
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...
Determine whether Rolle's Theorem can be applied to f on the closed interval [a,b]. (Select all that apply.)f (x) = sin(x), [0, 2π]If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f '(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.)I thought the derivative would be cos(x) so then cos(0) would be 1 but thatz wrong so now I don't understand...
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that 2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
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