How do you determine whether a function is concave or convex?

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).

How do you prove a function is concave?

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 know if an equation is convex?

If f and g are concave and a ≥ 0 and b ≥ 0 then the function h defined by h(x) = af(x) + bg(x) for all x is concave. If f and g are convex and a ≥ 0 and b ≥ 0 then the function h defined by h(x) = af(x) + bg(x) for all x is convex.

How do you show that a function is convex?

A function f : Rn → R is convex if and only if the function g : R → R given by g(t) = f(x + ty) is convex (as a univariate function) for all x in domain of f and all y ∈ Rn. (The domain of g here is all t for which x + ty is in the domain of f.)

How do you remember concave and convex?

The most important thing to remember is that concave means curving inwards and convex means curving outwards. A good tip is to focus on the ‘cave’ part of concave. If you remember that the mouth of a cave curves inwards, then you can remember that concave means bent inwards.

How do you prove a set is convex?

If C1 and C2 are convex sets, so is their intersection C1 ∩C2; in fact, if C is any collection of convex sets, then OC (the intersection of all of them) is convex. The proof is short: if x,y ∈ OC, then x,y ∈ C for each C ∈ C.

Does convex mean concave up or down?

Here’s a video by patrickJMT showing you how the second derivative test can tell us the concavity of a function. A function is concave up (or convex) if it bends upwards. A function is concave down (or just concave) if it bends downwards.

How do you know if a mirror is concave or convex?

Basically, the reflecting surface of convex mirror bulges outside while concave mirror’s bulges inwards. The major difference is the image that forms in these two mirrors. In other words, diminished images form in convex mirrors while enlarged images form in concave mirrors.

How do you know if a graph is convex or concave?

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). Consequently, how do you know if a function is convex?

How do you know if a function is convex or not?

If the matrix is: Positive-definite then your function is strictly convex. Positive semi-definite then your function is convex. A matrix is positive definite when all the eigenvalues are positive and semi-definite if all the eigenvalues are positive or zero-valued. Share Cite Follow answered Oct 1 ’20 at 0:19 Amogh MishraAmogh Mishra

How do you prove that a function is concave?

Now, compare the results with the above theorem which says that if the second derivative of a function is smaller than zero, then the function is concave. A concave function always bends upwards, so we can conclude that the function is concave and bends downwards.

How do you find the concavity and convexity of a curve?

A simple example to demonstrate concavity and convexity of a curve is with the help of a parabola.Consider y=x^2 . As you can see that this curve opens towards the positive x axis. So this curve is concave up or convex. On the other hand if we consider y=-x^2. This curve opens towards the negative y axis.