# Tagged Questions

21 views

### Proving that $f(x) = (1+x)^{-\theta}$ is convex using only definition

How can I prove that the function $f(x) = (1+x)^{-\theta}$ (where $\theta>0$ is a parameter) is convex using only definition, i.e. without using derivatives? I actually want to prove convexity of ...
49 views

308 views

### How to prove that the product of a decreasing monotonic function and a strictly increasing monotonic function is a concave function?

Given that I have a strictly increasing monotonic function $f$ and a decreasing monotonic function $g$, are there any nice properties to show that the product function $h(x) = f(x)g(x)$ is a concave ...
40 views

### Show this function is convex

Show that $f(x)=\frac{(\theta-x)\log_{x}\frac{x-\theta}{1-\theta}+x}{1-x}, ~x\in(\theta,\infty)$ is convex, where $\theta\in(0,1)$. $f(1)=\lim_{x\rightarrow 1}f(x)$. Numerical experiments suggest it ...
86 views

### Local Extrema and Global Extrema

When we have a convex function we know that a local minimum is a global minimum, and similarly for a concave function. What are some other situations where finding local extrema can yield global ...
696 views

### What are the main uses of Convex Functions?

Up till now I have just learned that the concept of convexity in functions of one variable is used to complete the graphs of functions, meaning to locate points of inflexion and see if the graph is ...