Suppose that a, b, and y are all functions.
If ay=by and y is one-to-one and onto, prove that a=b.
This looked very straightforward at first then I realised that a and b are functions, not values. I am therefore stuck on what to do.
Any help is much appreciated.
Suppose that f is continuous on the closed interval [a, b], a < b, and that f(a) =
A, f(b) = B, A < B.
Suppose further that f is strictly increasing on [a, b] (i.e. for any x1, x2 ∈ I, x1 <
x2, impliesf(x1) < f(x2)).
Let C be a number between A and B. Show that there is exactly one value of x in
(a, b) such that f(x) = C.
I used the bisection process to try to explain why this must be the case but i'm not convinced that's what they're looking for....any ideas???
I understand that if a sequence converges to a particular value then we call that sequence convergent.
However, what if the sequence converges to 2 values?
For example the sequence 2/3, -3/5, 4/7, -5/9 approaches -0.5 from below and approaches 0.5 from above. Do we just say that the sequence diverges?
Show that 5n+3 and 7n+4 are relatively prime for all n.
I'm fairly new at proofs and I think I've made decent inroads but can't seem to finish it elegantly.
Here is my attempt:
Show that gcd (5n+3 , 7n+4) = 1
let A=5n+3 and B=7n+4
if n is even then A is odd. If n is odd then B is odd. Therefore gcd cannot be even.
For an odd number (k) to divide A as well as B, the difference between A and B must be some multiple of k which can only be true if k equals 1.
That last line I don't think is sufficient but I don't know how to put it in more convincing terms. Help is appreciated.
Hey. I can find nothing on this material in my textbook and nothing online that's quite like it, so i'm hoping someone here can show me how to do this.
Consider the system
dx/dt = 4x + 5y + xy
du/dt = x + x^2 - y^2
Discuss the type and stability of the critical point at (0,0)