How many turning points can a function have?

Number of turning points in polynomial functions

• The most popular Functions are completely rational functions or Polynomial functions that are composed of power functions. The highest power indicates the degree of the polynomial. An example of such a function is this polynomial 3. Degree: f (x) = 2x³ - 5x² + 7.
• The second derivative f '' (x) of a function is responsible for the calculation of turning points. The zeros of this second derivative are possible x-values ​​of the turning point (if, in exceptional cases, they are not saddle points).
• So if you want to find out how many inflection points a polynomial has, you have to derive the polynomial twice and examine this function for zeros. If the polynomial has degree n, then the second derivative has degree n-2. The degree determines the maximum number of zeros, in this case n-2. A polynomial of the nth degree can therefore have a maximum of n-2 inflection points (but also fewer!).
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• In the example above, the second derivative has degree 1, so it is a linear function. This has a zero. A polynomial 3. Degree has a turning point (special case: f (x) = x³; there you have a saddle point at x = 0).

How many turning points do other functions have?

• Unfortunately, for all other possible functions one cannot establish such a simple, general rule as was the case for completely rational functions. But there are clues.


Third degree function - informative

Third degree functions are polynomials in which the variable x is ...

• Trigonometric functions like f (x) = sin x (and their extensions) are periodic. Here you can (if you do not limit yourself to a finite domain) calculate an infinite number of inflection points, since the course of the function repeats itself continuously.
• The exponential function f (x) = e^x as well as their inverse function, the natural logarithm f (x) = ln x, have no turning points, since both functions are constantly increasing.
• The root function f (x) = root (x), as an inverse function of the parabola, also has no point of inflection.
• So-called. broken rational functions of the form f (x) = g (x) / h (x), where g (x) and h (x) are polynomials, you have to use the second derivative to examine inflection points. There are no general rules as to how many turning points there are here.
• Also be careful with compound functions such as f (x) = -x² * e^x or f (x) = ln x / (x-1). These must also be examined using the second derivative.