Concave Function: Top Strategies to Solve

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

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
$lim_{n \to \infty} inf f (x_{n}) \geq f (x) .$

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 $z < z_{0}$ ?

Jefferson Booth 2022-11-10

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

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 $f$ )

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:
$\sum_{i = 1}^{N} a_{i} = A .$

Alexia Avila 2022-11-09

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

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 $f (a + c) - f (a) \leq f (b + c) - f (b)$ .

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?

Widersinnby7 2022-11-07

Let $I \subset R^{\geq 0}$ be an interval and let $f : I \to R^{\geq 0}$ be concave (and smooth enough). Weather the following inequality holds:
$f (a + b) \leq f (a) + f (b) .$
Is this true?

Jenny Schroeder 2022-11-07

What this means: Hence, v(6,000) < v(4,000) + v(2,000) and v(-6,000) > v(-4,000) + v(-2,000). These preferences are in accord with the hypothesis that the value function is concave for gains and convex for losses

Mark Rosales 2022-11-07

Let $f : R_{\geq 0} \to R_{\geq 0}$ , and $g : R_{\geq 0} \to R_{\geq 0}$ be two functions. Suppose that $f (0) > g (0) = 0$ , and $f$ is strictly increasing and concave, while $g$ is strictly increasing and strictly convex (so $f^{'} (x) > 0, f^{″} (x) \leq 0, g^{'} (x) > 0$ , and $g^{″} (x) > 0)$ ). Suppose that for some $x > 0$ we have that $f (x) - g (x) > 0$ . Is it true that then for every $y \in [0, x]$ it must be that $f (y) - g (y) \geq 0$ ?

Hallie Stanton 2022-11-06

Is maximum of increasing concave functions quasi-concave?

sbrigynt7b 2022-11-05

If a function $R \to R$ is strictly increasing and differentiable and it is true that $lim_{x \to \infty} f (x) = l$ where $l \in R$ , then $f$ is concave at some interval of the form $(k, + \infty)$ .

Laila Murphy 2022-11-05

Do $lim_{x ↓ 0} f (x)$ and $lim_{x ↑ 1} f (x)$ exist if $f$ is concave over $[0, 1]$ ?

Madison Costa 2022-11-05

Is this function:
$\exp (- | | A x | |^{2})$
concave in $A$ ??

dannigurl21ck2 2022-11-05

Explain this
The lagrangian being always concave because it is the pointwise infimum of a family of affine functions.

Kareem Mejia 2022-11-05

Suppose $f (x)$ is a function from $R$ to $R$ and $f$ is strictly quasi-concave. If $x^{}$ is a point such that $f^{'} (x^{}) = 0$ , then can we say that $x^{*}$ is a global maximum of this function? What about local maximum then?

Arendrogfkl 2022-11-05

Show that the log function logx is a concave function.

Demarion Ortega 2022-11-05

Show that the function:
$f (t) = - \log det Z - \sum_{i = 1}^{n} \log (1 + t λ_{i}) - trace ((Z + t V)^{- 1} Y)$
is concave in $t$ .

A concave function describes a curve bent inward, like the inside of a bowl. Examples include polynomials and logarithmic functions. To determine if a function is concave up or down, you can plot the equation on a graph or use derivatives. If the derivative is decreasing, the function is concave. If the derivative is increasing, it is convex. For more help with understanding concave functions, including equations and answers to specific problems, check out Plainmath many math tutors who are always available to help.

Calculus 1

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