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Let $f : R \to R$ be a concave function. Let $x_{0} \in R$ and $α > 1$ . Show that for all $x \in R$ ,
$f (x_{0} + x) + f (x_{0} - x) \geq f (x_{0} + α x) + f (x_{0} - α x) .$

Annie French 2022-11-11

What is a non-convex function? Is it a concave function?

ajakanvao 2022-11-10

How would one go about integrating the following?
$\int_{0}^{5} \frac{x}{x - 2} d x$

klasyvea 2022-11-10

Evaluate $\int \frac{\sin^{- 1} \sqrt{x}}{\sqrt{1 - x}} d x$

Emma Hobbs 2022-11-10

Let $X \subset R$ be convex, and let $f : X \to R$ be concave. Prove that for every $x \in X$ and every sequence ${x_{n}}$ converging to $x \in X$ , we have
$lim_{n \to \infty} inf f (x_{n}) \geq f (x) .$

tramolatzqvg 2022-11-10

How to solve these given?
$y = e^{\sqrt{x^{2} - 16}}$

nyle2k8431 2022-11-10

We have an inequality
$f (ϕ (z) + γ_{1} (z)) - f (ϕ (z) + γ_{2} (z)) < c,$
where $ϕ$ is decreasing in $z$ and $γ_{i}$ is increasing in $z$ for $i = 1, 2$ . All values are positive.How this inequality is preserved as we vary $z$ . Specifically, suppose this inequality holds for some $z_{0}$ . When will it also hold for any $z < z_{0}$ ?

inurbandojoa 2022-11-10

Evaluate the integral:
$\int_{- 1}^{1} \frac{x^{2}}{2} d x$

Jefferson Booth 2022-11-10

Concave function divided by a convex function. What is the result?

Alexia Avila 2022-11-09

Consider the increasing, concave function $x^{0.5}$ on $[0, 1]$ .

drzwiczkih5a 2022-11-09

Prove the following statement, in generic form, for $f > 0$ - concave function:
$A f (\frac{x}{A}) > a_{1} f (\frac{x}{a_{1}}) + a_{2} f (\frac{x}{a_{2}}) + \dots + a_{N} f (\frac{x}{a_{N}}),$
where $x > 0$ and:
$\sum_{i = 1}^{N} a_{i} = A .$

Paula Cameron 2022-11-09

Solve this integral:
$\int \frac{1}{16 + x^{2}} d x$

Siemensueqw 2022-11-09

Prove that $\frac{d}{d x} (\ln | x |) = \frac{1}{x}, x \neq 0$

Emma Hobbs 2022-11-09

Let $D$ be a convex set in $R^{n}$ and $f : D \to R$ a concave and $C^{1}$ function. How do I show that $x^{}$ is a global maximum for $f$ if and only if $f^{(1)} (x^{}) y \leq 0$ for all $y$ pointing into $D$ at $x^{*}$ (Here $f^{(1)}$ denotes the first derivative of $f$ )

Adrian Brown 2022-11-09

Find the first derivative of the following exponential and logarithmic functions
$a) y = e^{5 x} b) y = x e^{2 x} c) y = 2^{(x - 1)}$

Alberto Calhoun 2022-11-08

Prove: $f : [0, \infty) ⟶ [0, \infty)$ that is concave and $a > b > 0$ . Then given a constant $c > 0$ I claim that $f (a + c) - f (a) \leq f (b + c) - f (b)$ .

tramolatzqvg 2022-11-08

I was wondering if someone could help me find the definite integral of this:
$\int_{R 1}^{R 2} \frac{t d t}{(t^{2} + K^{2})^{3 / 2}}$

kituoti126 2022-11-08

IProve that exponential functions $a^{n}$ have different orders of growth for different values of base a>0.
It looks obvious that when a=3 it grows faster when compared to a=2. But how do i make a formal proof for this? Thanks for your help.

MISA6zh 2022-11-08

Suppose we have two functions, $f (x), g (x)$ , and $f (x)$ is convex and $g (x)$ is concave. What are some conditions that will guarantee that $f (x) g (x)$ is concave?

Adrian Brown 2022-11-08

Each logarithm should involve only one variable and should not have any exponents. Assume that all variables are positive.
$\log (z y^{3}) = ?$

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