In this lecture we return back to functions of one variable, defined on an open or closed interval on the real axis.
Definition. A function is called twice (two times) differentiable, if it is differentiable at all points of the segment , and the derivative considered as a scalar (numeric) function on this segment, is also differentiable.
Iterating this construction, we say that a function is times differentiable, if it is differentiable and its derivative is times differentiable. This is an inductive definition.
Variations. Sometimes it is convenient to say that a function is 0 times differentiable, if it is continuous on the segment. If this agreement is used as the base of induction, then this would define a slightly more restricted classes of times differentiable functions, usually denoted by .
- Give non-polynomial example of a function which is infinitely many times differentiable.
- Give example of a function that has exactly 7 derivatives, but not 8, on the segment .
As was already established, existence of the first derivative allows to construct the linear approximation for a given function. If this approximation is not degenerate (i.e., the derivative is non-vanishing), it allows to study the extremal properties of the function, in particular,
- Guarantee a certain type of the extremum at the endpoints of the segment;
- Guarantee the absence of extremum at the interior points of the interval .
It turns out that higher order derivatives allow to construct approximation of functions by polynomials of higher order, and this approximation sometimes guarantees presence/type or absence of extremum. In cases it does not, one needs more of the same.
Theorem. If a function is times differentiable on a segment containing the origin , then there exists a unique polynomial of degree which gives an approximation of at the origin with the accuracy ,
The coefficients of this polynomial are proportional to the respective higher derivatives at the origin, .
Remark. There is nothing mysterious in the formulas: if is a polynomial itself, then the higher order derivatives can be easily computed for each monomial separately:
This proves the formulas for the coefficients via the derivatives of the polynomial.
For the proof we need the following lemma, which is very much like the intermediate value theorem.
Lemma (on finite differences). For a function differentiable on the interval the normalized finite difference coincides with the derivative at some intermediate point .
Proof of the Lemma. Consider the auxiliary function . Then and the same finite difference for this function is equal to zero. Since the function takes equal values at the endpoints, either its maximum or its minimum is achieved at some point inside the interval . By the Fermat rule, . By construction,.
Proof of the Theorem. The formulas for the coefficients imply that the difference is a function which is times differentiable and all its derivatives vanish at the origin. We have to prove that .
For this is obvious: by definition of differentiability, for function with zero derivative we have . Reasoning by induction, consider the function and its derivative . By the inductive assumption, for an arbitrary on a sufficiently small interval.
By the Lemma, for some . Using the inequality , we conclude that . Since can be arbitrary small, we conclude that the limit is smaller than any number, i.e., must be zero.
Definition. The polynomial above is called the Taylor polynomial of order for the function at the origin . One can easily modify this to become the definition if the Taylor polynomial at an arbitrary point .
Application to the investigation of functions
Any times differentiable function can be written (near the origin, but this is not a restriction!) as its Taylor polynomial of degree plus an error term which is fast decreasing (faster than the highest degree term of the polynomial). Hence this term cannot affect the extremal properties of the polynomial. In particular, if and the Taylor polynomial has a minimum (maximum) at , so does the function itself.For quadratic polynomials the horns of the parabola go up (resp., down) if its principal coefficient is positive (resp., negative). This immediately proves the following result.
Second-order sufficient condition.If is a twice differentiable function and is a critical point, , then the following holds:
- If , then is a local minimum,
- If , then is a local maximum,
- If , everything depends on the Taylor polynomial of degree 3.
Problem. Find conditions for a degree 3 polynomial to have a local maximum/minimum on (a) interval , (b) semi-interval . Formulate the third order necessary/sufficient conditions for extremum in the interior point (resp., left/right endpoint) of the domain of a non-polynomial function .