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Aug 16, 2019 · What you gave is the standard definition of a convex function. If $f$ is supposed to be continuous, it is enough to check that $$f\left(\frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2}$$ for all $x,y$. If $f$ is twice differentiable, it is enough to check that the second derivative is non negative.
Prove that each of the following functions is convex on the given domain: f (x)=e^ {b x}, x \in \mathbb {R}, where b is a constant. f (x)=x^ {k}, x \in [0, \infty) and k \geq 1 is a constant. f (x)=-\ln (1-x), x \in (-\infty, 1). f (x)=-\ln \left (\frac {e^ {x}} {1+e^ {x}}\right), x \in \mathbb {R}.
Oct 3, 2018 · $$f\text{ is convex} \implies \operatorname{epi}(f)\text{ is convex}$$ Since it is known that $$f\text{ is convex} \iff \operatorname{epi}(f)\text{ is convex},$$ I would like to prove $$f\text{ is convex} \impliedby \operatorname{epi}(f)\text{ is convex}$$ specifically. Current attempt. Suppose $\operatorname{epi}(f)$ is convex, and that we have:
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.
Nov 2, 2020 · Prove that a function of $n$ variables is concave if and only if the set below its graph in $\mathbb{R}^{\mathbf{n+1}}$ is a convex set.
5 days ago · A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.
The function f is weakly convex if, for every a; b 2 I and every t 2 (0; 1), Terminology: Spivak's \convex" means strictly convex. Many other author's \convex" means weakly convex. Example: the function f(x) = 1=x is convex on (0; 1) and concave on (1 ; 0). Geometric meaning: f is weakly convex i.
