Question

Can someone help me?. I am having trouble understanding exactly what concavity and convexity of a function really mean.

I,ve found out that the function `y=x^2` is said to be convex, but when you are faced with functions like `y=x^3` this changes and I am not sure what is convex and what is concave. Also, when the second derivative is `0` and you test either side of this point, does the sign change if it's a point of inflexion.

Answer 1

See explanation

Explanation:

The term concavity refers to the rate of change of the derivative of the function; in other words, the second derivative of the function.

Graphically, a function is convex (i.e concave upwards) on the interval `(a,b)` if all the points on a straight line between a and b are above the function on that interval. Graphing the `x^2` function, we see this is the case everywhere along the function. It is also the case for `x^3` on `(0,oo)`

A function is concave, i.e. concave downwards, on an interval if all the points on the aforementioned line are below the function on that interval. This is the case for `x^3` on the interval `(-oo,0)`.

Mathematically, a function is convex when its second derivative is greater than 0, and concave when its second derivative is less than 0. If the second derivative is equal to 0, then it is an inflection point if the concavity changes when you cross that point, i.e. if the second derivative changes from negative to positive or positive to negative.