Question

Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i) Prove that if y > 0, then there is a solution x to the equation f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove that the function f : R → R is strictly monotone. (iii) By (i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why the derivative of the inverse function, (f^ −1 )'(y) exists, for all y ∈ R, and find its value at y = 0.

Answer #1

Let f : [0,∞) → [0,∞) be defined by, f(x) := √ x for all x ∈
[0,∞), g : [0,∞) → R be defined by, g(x) := √ x for all x ∈ [0,∞)
and h : [0,∞) → [0,∞) be defined by h(x) := x 2 for each x ∈ [0,∞).
For each of the following (i) state whether the function is defined
- if it is then; (ii) state its domain; (iii) state its codomain;
(iv) state...

Prove the following theorem:
Theorem. Let a ∈ R
and let f be a function defined on an
interval centred at a.
IF f is continuous at a
and f(a) > 0 THEN
f is strictly positive on some interval
centred at a.

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable.
Then f and g differ by a constant if and only if f ' (x) = g ' (x)
for all x ∈ [a, b].
b) For c > 0, prove that the following equation does not have
two solutions. x3− 3x + c = 0, 0 < x < 1
c) Let f : [a, b] → R be a differentiable function...

Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...

Prove: If D = Q\{3}. R = Q\{-3}, and f:D-> R is defined by
f(x) = 1+3x/3-x for all x in D, then f is one-to-one and onto.

Consider the function f defined on R by f(x) = ?0 if x ≤ 0, f(x)
= e^(−1/x^2) if x > 0.
Prove that f is indefinitely differentiable on R, and that
f(n)(0) = 0 for all n ≥ 1. Conclude that f does not have a
converging power series expansion En=0 to ∞[an*x^n] for x near the
origin. [Note: This problem illustrates an enormous difference
between the notions of real-differentiability and
complex-differentiability.]

The function f(x) = 3x 4 − 4x 3 + 12 is defined for all real
numbers. Where is the function f(x) decreasing?
(a) (1,∞) (b) (−∞, 1) (c) (0, 1) (d) Everywhere (e) Nowhere

Let f : R − {−1} →R be defined by f(x)=2x/(x+1).
(a)Prove that f is injective.
(b)Show that f is not surjective.

Let f : R → R be a bounded differentiable function. Prove that
for all ε > 0 there exists c ∈ R such that |f′(c)| < ε.

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