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A function $f : A \to A$ has convex/concave potential if there is $F : R^{n} \to R$ such that $\nabla F = f$ , and $F$ is convex/concave.
Let $A \subset R^{n}$ be a compact set.
Are there efficient ways to generate not trivial examples of such functions such that $f : A \to A$ is onto $A$ ?

raapjeqp 2022-10-31

Does $\begin{aligned} lim_{n \to \infty} \int_{a}^{b} f (x)^{p_{n} (x)} d x = \int_{a}^{b} f (x)^{p (x)} d x \end{aligned}$ hold in general?

Nigro6f 2022-10-31

Is $\frac{1}{T r (X^{T} P X)}$ concave, if $P$ is p.s.d?

JetssheetaDumcb 2022-10-31

Concave function - Generic proof
$f (x) \leq 2 f (\frac{x}{2}) \leq 3 f (\frac{x}{3}) \leq \dots \leq N f (\frac{x}{N})$

Alexander Lewis 2022-10-31

Consider two function $f (x)$ and $g (x)$ defined on $[0, 1]$ and $f (0) = g (0) = 0$ and $f (1) = g (1) = 1$ .
Prove that the semi-elasticity of $g$ is greater than the semi-elasticity of $f$ for all $x \in (0, 1)$ ?

beefypy 2022-10-31

Let $f, g : [0, 1] \mapsto [0, 1]$ becontinuously increasing concave functions such that $f (0) = g (0) = 0$ and $f (1) = g (1) = 1$ . What is the maximum intersection points between $f$ ad $g$ .

rhenan5v 2022-10-30

Consider $g \in C^{1} [0, 1]$ . We say that $g$ is nowhere convex (concave, resp.) on $[0, 1]$ if there is no open interval $I \subseteq [0, 1]$ on which $g$ is convex (concave, resp.) Is it possible to find a function $g$ which satisfies:
$g$ is $C^{2} [0, 1]$ and $g$ is nowhere convex and nowhere concave on $[0, 1]$ ?

princetonaqo3 2022-10-30

1) Find the limit
$lim_{x \to 1} \log_{5} (3 x + 6)^{2}$
2) Find the limit
$lim_{x \to 7} - 3 \ln e^{x}$

miklintisyt 2022-10-30

Find:
$\int_{1}^{5} \frac{d x}{3 x + 1}$

Paloma Sanford 2022-10-30

How to integrate $\int \frac{\sin^{2} θ}{\cos^{5} θ} d θ$

Cyrus Travis 2022-10-30

How to show $lim_{x \to - N} (x + N) Γ (x)$

caritatsjq 2022-10-30

Is the set of all concave functions a convex set?

link223mh 2022-10-29

Consider the following non linear equation where x is the independent variable, y is the dependent variabie, and A and e are constants.
$y = - A x^{B}$
Which of the following is the resulting gradient variable in linearised form?
Select one:
$a) l o g 10 (B) b) A c) e x p (B) d) B e) l o g (- A)$

Francis Oliver 2022-10-29

Approximate the function at the value of x to four decimal places.
$h (x) = \log_{4} x - 2 \log_{8} 1.2 x, x = 5.3$

princetonaqo3 2022-10-29

Consider a function $f : [0, 1] \to R^{+}$ such that $f (0) = 0$ and $f (x) \leq f (y)$ for all $x \leq y$ (i.e f is monotone).Is a positive, monotone and sub-additive function concave?

Trace Glass 2022-10-29

Is there a nice way to integrate this?
$\underset{x^{2} + y^{2} + z^{2} ⩽ 1}{∭} x^{2 n} + y^{2 n} + z^{2 n} d V$

Trace Glass 2022-10-29

Let $f$ be a nonnegative concave function on a convex domain $D$ . Assume that $f$ achieves a positive value at some point $x \in D$ . Then $f (y) > 0$ for all $y \in D^{o}$ .

racmanovcf 2022-10-29

Suppose we know the definition of rational exponents of positive real numbers. If we define the exponential function $y = a^{x} (a > 0)$ by extending rational exponents to real exponents by the following limiting process:
$a^{x} = lim_{r_{i} (\in Q) \to x} a^{r_{i}},$
then how can we prove the exponential function is continuous in R?

Kayla Mcdowell 2022-10-29

Suppose $f$ and $: [0, l) \to R$ are continuous, concave and increasing where $l < \infty$ . Can we claim that they intersect at most finitely many points? What if we replace l with $\infty$ ?

racmanovcf 2022-10-29

Let $g : R_{+} \to R$ , defined as follows : $g (x) = \frac{x}{1 + x}$ , it's concave ( since $g^{″} (x) \leq 0$ ).
Now we define the sup-norm on $C^{\infty} [0, 1]$ , $ψ (f) = sup_{x \in [0, 1]} | f (x) |$ .
prove that $g o ψ$ is concave.

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