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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.
- real analysis - Prove that every convex function is ...
A function f: (a, b) → R is said to be convex if f(λx + (1 −...
- real analysis - Prove that every convex function is ...
Prove that \(cf\), \(f + g\), and \(\max \{f, g\}\) are convex functions on \(I\), where \(c \geq 0\) is a constant. Find two convex functions \(f\) and \(g\) on an interval \(I\) such that \(f \cdot g\) is not convex.
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.
A function f: (a, b) → R is said to be convex if f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y) whenever a <x, y <b and 0 <λ <1. Prove that every convex function is continuous. Usually it uses the fact: If a <s <t <u <b then f(t) − f(s) t − s ≤ f(u) − f(s) u − s ≤ f(u) − f(t) u − t.
A function f: Rn!Ris convex if and only if the function g: R!Rgiven by g(t) = f(x+ ty) is convex (as a univariate function) for all xin domain of f and all y2R n .
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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 proof for concave functions follows by the following result ; If \(g\) is concave, \(-g\) is convex. Note that a function is convex if and only if the following holds : \( f(\mu x + (1-\mu) y) \leq \mu f(x) + (1-\mu) f(y) \quad \forall \, \, \mu \in (0,1) ; \, x,y \in I ;\, x < y \)
