10) (4 points) Prove or disapprove the function f:R → R such f(x) = 3x -...
Which function below is the inverse of f:R-{2} → R-{3} ut of f(x)= -3x+1 X X-2 Select one: O a. f-1: R-{3} → R-{2} f'(x)=2x+1 X-3 O b. f-1: R - {2} → R-{3} F"(x) = 2X+1 X-3 f-:R-{2} → R-{3} f(x)= x-2 3x + 1 O d. f-1: R - {3} → R-{2} ... X-2 hook....pdf - POS Week 171 ..hantal ob. F-R-{2} → R-{3} F-1(x)=2x+1 3 F-1R- {2} → R - {3} X-2 pe d. f":R-{3} → R...
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
(6) Let fel ), where is Lebesgue measure on R. Define F:R → R by F(x) = f' f(t) dx. (a) Prove that F is a continuous function. (b) Prove that F is uniformly continuous on R. (Note that R is not compact.)
2) (12) f:R-(3/2)-R-10, (x) 1/(3 2x) g:R--21->R-1o), g (x)1/ (x 2) h:R-(-4/3]-R-(1/3), h(x) (f o g) (x) Verify if h(x) is one to one and onto. If it is, find the inverse function of h(x). 2) (12) f:R-(3/2)-R-10, (x) 1/(3 2x) g:R--21->R-1o), g (x)1/ (x 2) h:R-(-4/3]-R-(1/3), h(x) (f o g) (x) Verify if h(x) is one to one and onto. If it is, find the inverse function of h(x).
Consider the function f:R + R defined by if x is rational f(x) = if x is irrational. Find all c € R at which f is continuous. C
Exercise 7.9. Assume f:R → R. (a) Let t € (1,0). Prove that if |f(x) = alt for all x, then f is differentiable at 0. (b) Let t € (0,1). Prove that if f(x) = |x|* for all x, and f(0) = 0, then f is not differentiable at 0. (c) Give a pair of examples showing that if |f(x)= |x|for all I, then either conclusion is possible.
Let f:R → Z defined by f(x) = 23 – 2. Prove that f is a one-to-one correspondence (i.e., a bijection).
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...
Is x = 0 a relative extrema for function f : R → R that is given by f(x) = sinx − cosx − 1/2(1 + x)^2 Question 3. Is x = 0 a relative extrema for function f:R + R that is given by 1 COS X f(x) = sin x (1 + x)? 2 Prove your claim. State any theorem that is applied in your proof.