All Questions
7
questions
1
vote
1
answer
34
views
A claim regarding some summations of monotonic functions in fraction
I am trying to prove this claim but it seems the math somehow does not work out...
Let $f_1,f_2,g_1$ and $g_2$ be real-valued, strictly positive, continuously differentiable and strictly decreasing ...
- 111
4
votes
6
answers
682
views
Is $x^3$ really an increasing function for all intervals?
I had an argument with my maths teacher today...
He says, along with another classmate of mine that $x^3$ is increasing for all intervals. I argue that it isn't.
If we look at conditions for ...
- 45
0
votes
1
answer
26
views
Is it possible for a function to be continuously derivable over its entire open domain except for a removable discontinuity?
For example, is there a function $f \in \mathcal{C}(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R} \setminus \{ 0 \})$ such that
$$ \exists \lim_{x \to 0} f'(x) = \lim_{x \to 0} \lim_{y \to x} \frac{f(y)-f(...
- 33
2
votes
1
answer
102
views
Is it possible to find the local maximum of $\sqrt[x]{x}$ without using derivative?
Let $f(x)=\sqrt[x]{x}$, where $x\in\mathbb{R^{+}}$
Using derivative,
$$\frac{d}{dx}(x^{\frac 1x}) = -x^{\frac 1x - 2} (\log(x) - 1)$$
$$f'(x)=0 \longrightarrow x=e$$
$$\text{max}\left\{\sqrt[x]{x}...
user548054
1
vote
3
answers
210
views
Everywhere continuous and differentiable $f : \mathbb{R} → \mathbb{R}$ that is not smooth?
I can't seem to find any counterexamples to the statement "all functions that are continuous and differentiable at every point of the reals are smooth," nor can I find anyone asserting or proving this ...
- 23
7
votes
2
answers
810
views
Link between polynomial and derivative of polynomial
I can't seem to solve this problem, can anyone help me please? The problem is:
Let real numbers $a$,$b$ and $c$, with $a ≤ b ≤ c$ be the 3 roots of the polynomial $p(x)=x^3 + qx^2 + rx + s$. Show ...
- 73
3
votes
1
answer
1k
views
Rudin's definition of derivative
Walter Rudin's Principle of Mathematical Analysis defines the derivative as follows in Definition 5.1:
Let $f$ be defined (and real-valued) on $[a,b]$. For any $x \in [a,b]$ form the quotient
$$\...
- 1,353