Many important properties of functions, such as monotonicity, extreme point, concavity and convexity, are expressed by the relationship between function increment and independent variable increment. The differential mean value theorem (Lagrange mean value theorem and Cauchy mean value theorem) just establishes the relationship among function increment, independent variable and derivative. Therefore, according to it, the monotonicity, extreme point, concavity and inflection point of the function can be discussed by derivative.

On the basis of understanding relevant theorems, we can master the methods of judging monotonicity and concavity of functions, and the methods of finding extreme values and inflection points with derivatives, which are embodied in the drawing of functions (including finding asymptotes of functions).

Another important application of differential calculus is to find the maximum and minimum of a function. Master the method of finding the best value and solve simple application problems. The key to finding the maximum value is to find the stagnation point.

Extended data:

Differential mean value theorem, Cauchy theorem:

If the functions f(x) and F(x) satisfy

(1) is continuous on the closed interval [a, b];

(2) Derivable in the open interval (a, b);

(3) For any x∈(a, b), F'(x)≠0.

Then there is at least one point ξ in (a, b), so that the equation [f (b)-f (a)]/[f (b)-f (a)] = f' (ξ)/f' (ξ) holds.

[Mean value theorem] is divided into:? Differential mean value theorem and integral mean value theorem;

The above three are differential mean value theorems. The first mean value theorem of definite integral is:

The definite integral of f(x) on a to b is equal to f(ξ)(b-a) (ξ∈[a, b] makes this formula hold).

Note: The integral mean value theorem can be deduced according to the mean value theorem, so the same ξ∈[a, b] are all closed intervals.