How do you determine if a function is convex or concave two variables?
How do you determine if a function is convex or concave two variables?
For a twice-differentiable function f, if the second derivative, f ”(x), is positive (or, if the acceleration is positive), then the graph is convex (or concave upward); if the second derivative is negative, then the graph is concave (or concave downward).
Is concave function differentiable?
1. A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. Points where concavity changes (between concave and convex) are inflection points.
Is a concave function always continuous?
This alternative proof that a concave function is continuous on the relative interior of its domain first shows that it is bounded on small open sets, then from boundedness and concavity, derives continuity. If f : C → R is concave, C ⊂ Rl convex with non-empty interior, then f is continuous on int(C).
How do you show a function is concave?
To find out if it is concave or convex, look at the second derivative. If the result is positive, it is convex. If it is negative, then it is concave. To find the second derivative, we repeat the process using as our expression.
What is the convex concave rule?
The concave-convex rule states that if a concave surface moves on a convex surface, roll and slide must occur in the same direction, and if a convex surface moves on a concave surface, roll and slide occurs in opposite directions.
What is concave curve?
Concave describes an inward curve; its opposite, convex, describes a curve that bulges outward. They are used to describe gentle, subtle curves, like the kinds found in mirrors or lenses. If you want to describe a bowl, you might say there is a large blue spot on the center of the concave side.
Is linear function concave?
A linear function will be both convex and concave since it satisfies both inequalities (A. 1) and (A. 2). A function may be convex within a region and concave elsewhere.
Is every convex function is continuous?
Since in general convex functions are not continuous nor are they necessarily continuous when defined on open sets in topological vector spaces. But every convex function on the reals is lower semicontinuous on the relative interior of its effective domain, which equals the domain of definition in this case.
Is log x a concave function?
The logarithm f(x) = log x is concave on the interval 0
Can a function be both concave and convex?
That is, a function is both concave and convex if and only if it is linear (or, more properly, affine), taking the form f(x) = α + βx for all x, for some constants α and β. Economists often assume that a firm’s production function is increasing and concave.
What is the concave rule?
Is a bowl concave or convex?
An object with a concave shape curves inward, such as a spoon or a bowl. The middle is thinner than the edges. An object with a convex shape is one that curves outward, such as a basketball or a baseball.
When is a function of many variables concave?
Let f be a function of many variables, defined on a convex set S. We say that f is concave if the line segment joining any two points on the graph of f is never above the graph; f is convex if the line segment joining any two points on the graph is never below the graph.
When is a twice differentiable function f concave?
A twice-differentiable function f of a single variable defined on the interval I is concave if and only if f”(x) ≤ 0 for all x in the interior of I convex if and only if f”(x) ≥ 0 for all x in the interior of I.
Is the graph of a concave function symmetric?
The generalization of this result to concave functions of many variables says that the graph of such a function lies everywhere on or below all of its tangent planes. As for a function of a single variable, a symmetric result holds for convex functions.
How to find where f is concave in multivariable calculus?
To find where f is concave, we determine where − f is convex, which in turn implies − H is positive semidefinite. This only reverses the first inequality, and we obtain the same result, that f is not concave anywhere. Thanks for contributing an answer to Mathematics Stack Exchange!