convex-analysisstatistics
I'm learning concave/convex functions. Wikipedia article provides us with example functions of concave or convex.
However, it does not provide strictly concave function.
Do you have some examples of strictly concave function?
It would be very helpful if you give me the reason how strictly concavity property is used in some analysis.
The second is neither convex nor concave - that's easy to determine simply by looking at it. Along the line $y=x$, it is convex as a 1D function; along the line $y=-x$ it is concave. It is neither quasi-convex nor quasi-concave: to show not quasi-concave, consider the points $x = (0, 1)$, $y = (-1, 0)$, so $f(x) = f(y) = 0$. Parametrise the function along that line segment by $\lambda$; then $f(\lambda) = \lambda (\lambda - 1) < 0 = \min \{ f(x), f(y) \}$. You can rotate to get non-quasi-convexity.
The first is convex but not concave, and it's not quasi-concave. Examine the value of $f$ at the points $x=1/3, x=10, x=1$ to see that it's not quasi-concave. It's convex again by inspection or by showing that its second derivative is strictly positive.
I don't know why I couldn't see it initially but for any given $\lambda$ we have an affine function, x being constant so the basic idea is (just looking at the graph):
wherein the "x-axis" is lambda and the "y-axis" is the value of the dual function g($\lambda$)
Best Answer
For any strictly convex function $f$, the function $-f$ will be strictly concave.
An example would be $$f(x)=-x^2$$ for $x\in\mathbb R$.