How do you know if concavity is strict?
Functions of a single variable If f is twice-differentiable, then f is concave if and only if f ′′ is non-positive (or, informally, if the “acceleration” is non-positive). If its second derivative is negative then it is strictly concave, but the converse is not true, as shown by f(x) = −x4.
How do you test concavity?
To find when a function is concave, you must first take the 2nd derivative, then set it equal to 0, and then find between which zero values the function is negative. Now test values on all sides of these to find when the function is negative, and therefore decreasing.
How do you tell if something is concave up or down?
Taking the second derivative actually tells us if the slope continually increases or decreases.
- When the second derivative is positive, the function is concave upward.
- When the second derivative is negative, the function is concave downward.
How do you find concave up and down Examples?
If f′′(x)<0, the graph is concave down (or just concave) at that value of x. If f′′(x)=0 and the concavity of the graph changes (from up to down or vice versa), then the graph is at an inflection point. Determining concavity obviously requires finding the second derivative, if it even exists.
When does a function need to be quasi concave?
IF the function is monotonic, on a real interval, then the function will be quasi convex and quasi concave, that is a sufficient condition, although not necessary for the function to be quasi linear ( both quasi convex or quasi concave) so if the derivative
Which is weaker the notion of concavity or quasiconcavity?
The notion of quasiconcavity is weaker than the notion of concavity, in the sense that every concave function is quasiconcave. Similarly, every convex function is quasiconvex. First note that the set S on which f is defined is convex, so we have (1 − λ)x + λy ∈ S, and thus f is defined at the point (1 − λ)x + λy.
Is the contour of a mountain convex or quasiconcave?
A function with the property that for every value of a the set of points ( x , y) such that f ( x , y ) ≥ a —the set of points inside every contour on a topographic map—is convex is said to be quasiconcave . Not every mountain has this property. In fact, if you take a look at a few maps, you’ll see that almost no mountain does.
When is the condition of quasiconcavity satisfied?
It is if every straight line connecting two points on the surface lies everywhere on or under the surface. If, for example, the mountain is a perfect dome (half of a sphere), then this condition is satisfied, so that the function defined by its surface is concave. The condition is satisfied also if the mountain is a perfect cone.