Timeline for Convexity of the natural exponential fuction - directly from the definition
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 13, 2016 at 3:17 | answer | added | user232552 | timeline score: 1 | |
| Jan 12, 2016 at 22:56 | comment | added | user74973 | @copper.hat Often, I am pleasantly surprised with the responses. | |
| Jan 12, 2016 at 22:47 | comment | added | copper.hat | @user74973: It is good to ask such questions. | |
| Jan 12, 2016 at 22:43 | comment | added | user74973 | @copper.hat Ha! Ha! I understand that this is a contrived, pedagogical post. As suggested in my previous comment, the convexity of some functions can be shown via some algebra and the convexity of other basic functions. | |
| Jan 12, 2016 at 22:30 | comment | added | copper.hat | As Galbraith said to Kennedy, "treating exponentials without derivatives is like fornicating through a mattress". | |
| Jan 12, 2016 at 22:27 | comment | added | user74973 | @copper.hat Maybe I should have started with the phrase "without differentiation." I am curious whether this can be shown using algebraic properties of inequalities or using various inequalities. For example, see the argument that Martin R gave for the post "Inequality of a weighted mean of $x^{k}$ and $y^{k}$." | |
| Jan 12, 2016 at 22:09 | comment | added | copper.hat | Here is another approach that might play into limits: Use the cfollowing characterisation of convex functions proofwiki.org/wiki/… and show that $\exp$ is its own derivative and positive everywhere. | |
| Jan 12, 2016 at 21:35 | comment | added | user74973 | @Ian Pedagogical. I am writing notes for a real analysis course (Calculus course), and I wanted to discuss this elementary function right after discussing limits. | |
| Jan 12, 2016 at 21:32 | comment | added | Ian | Sorry, missed that part. Why do you not want to proceed this way, though? | |
| Jan 12, 2016 at 21:31 | comment | added | user74973 | @Ian Read my post again. I started it with the phrase "without using the Second Derivative Test." | |
| Jan 12, 2016 at 21:27 | comment | added | Ian | I would probably proceed using the lemma that a $C^2$ function with a nonnegative second derivative is convex. That fact is easy to prove in numerous ways for the exponential function. | |
| Jan 12, 2016 at 21:22 | history | asked | user74973 | CC BY-SA 3.0 |