How to prove that convex function has an increasing slope?
2015年2月9日 · One possible way of describing the slope of this line is $\frac{f(x_3)-f(x_1)}{x_3-x_1}$. However, it is straightforward to show that for any two points $\left(a,f(a) \right), \left(b,f(b) \right)$ that are part of this line, the slope can be equivalently described as $\frac{f(b)-f(a)}{b-a}=\frac{f(x_3)-f(x_1)}{x_3-x_1}$ .
How can we prove that slopes increase in a convex function
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Intuition behind convex functions - Mathematics Stack Exchange
(3), being very similar to the definition of a derivative, is the only one that makes sense to me, that is: a function is convex if the slope between points $(u,f(u))$ and $(v,f(v))$ is lesser than the slope between $(v,f(v))$ and $(w,f(w))$. (2) seems to look at areas of rectangles, however, that is about everything I could say about it.
calculus - Do I have the correct visual intuition for convex
2023年5月8日 · A convex function must be continuous, and $|x|$ is continuous at $0$, but it is not differentiable at $0$ and therefore the slope is undefined there. The intuition is fine, but you have to keep in mind it's just that, an intuition, not a substitute …
(Strictly) concave/convex function: Increasing / decreasing slope …
2014年8月2日 · Is the power of a positive strictly monotonic function convex/concave? 1 Quasi-concavity of sum of strictly increasing concave and strictly decreasing convex function.
How should the contour lines of a convex function look?
2018年8月20日 · My book asks me to sketch the contour lines of a convex function and discuss why it looks like this. I've been thinking about this for a while. My guess is that each contour would have to be convex too, since if it's not, then I found a place of intersection with the shape where a line between two points of itself don't lie in the shape.
real analysis - Properties of increasingly convex function ...
2016年1月9日 · Convex functions and its differential properties [Zorich's book] Hot Network Questions Why is Young's modulus represented as a single value in DFT calculations?
What is concave slope? - Answers
2024年6月1日 · What are concave and convex slopes? A concave slope curves inward or has a depression in the middle, resembling a bowl shape. In contrast, a convex slope curves outward, forming a mound or hill shape.
analysis - Convexity of piecewise constant functions - Mathematics ...
2021年6月6日 · Similarly, functions of constant slope are convex; they are not strictly convex. (Note that wiki's definition consistently allows for equality in all their inequalities.) $\endgroup$ – Jacob Manaker
convex analysis - Definition of convexity - Mathematics Stack …
In this case, we may define convex by saying all points on such segments lie in the set. Derived from that, a function is described as convex when the set of points above (or maybe below) of its graph is convex. Note that a function may be convex …