1. mean value theorem
There are Fermat Lemma, Rolle theorem and Lagrange mean value theorem in differential calculus.
Lagrange Theorem If the function satisfies:
(i) In a closed interval, it is upper continuous;
(ii) In the open interval, it is internally derivable,
Then at least one point is there, so
or
It is easy to understand from Figure 3 that when the function satisfies (i) and (ii), that is, it is a continuous curve, and each point has a tangent, then there is at least one point p on the curve (as long as the chord AB moves in parallel), so that the tangent of the curve at this point is parallel to the chord AB, that is, the slope of the tangent at point P is equal to the slope of the chord AB.
It should be noted that Lagrange theorem does not give a specific method of evaluation, it only affirms the existence of values, and there is at least one. As shown in the function in Figure 3, in, there are and two. The significance of Lagrange's theorem is to establish the relationship between the change of function in the interval and the derivative of function at a certain point in the interval, thus providing a theoretical basis for studying the properties of function in the interval with derivative.
2. Study the properties of functions with derivatives.
For the convenience of discussion, we will use the symbols and, which represent the open interval and the closed interval respectively.
Now we use derivatives to study the monotonicity of functions. Let the function be continuously differentiable in the domain. If the function is monotonically increasing in the graph, then its graph is a curve rising along the axis, as shown in Figure (a), when the tangent slope of each point on the curve is greater than or equal to zero (); If the function monotonically decreases in the graph, then its graph is a curve that decreases along the axis, as shown in Figure (b), when the tangent slope of each point on the curve is less than or equal to zero (). Therefore, the monotonicity of a function is closely related to the sign of its derivative.
Conversely, can the sign of the derivative be used to judge the monotonicity of the function?
Sign of the first derivative
Take two points, including
(& lt& lt)
There is the above formula, if, there is, so the interval is monotonically increasing. Similarly, it can also be interpreted as interval monotony decreasing.
From this, the method of judging monotonicity of function can be summarized.
If the interval is continuous and differentiable, then
(1) If the function is satisfied in the interval, then the function is increasing function in the interval;
(2) If the function is satisfied in the interval, the function is a decreasing function in the interval.
(3) If the function is satisfied in the interval, the function is unchanged in the interval.
In addition, the absolute value of the derivative tells us the magnitude of the rate of change. The greater the absolute value, the steeper the function curve; When the absolute value is small, the function curve is flat. With this in mind, you can judge some properties of a function from its derivative.
Concave degree of curve
If it is differentiable in an interval, the first derivative tells us that if it is in an interval, it will increase in the interval;
If it is in a certain interval, it is decreasing in this interval.
If it increases in a certain interval, its function curve bends upward or is called concave; If it decreases in a certain range, its function curve bends downward or is called concave. When bending upward, the slope of the tangent of the curve increases with the increase, as shown in the figure; When bending downward, the slope of the tangent of the curve decreases with the increase.
The point is the inflection point of the function, that is, the function curve is concave upward on the left side of the point and concave downward on the right side of the point in the region, which is the dividing point of the curve from concave upward to concave downward.
Sign of the second derivative
The concavity or concavity of the function curve and the inflection point of the curve can be determined by the second derivative of the function.
If the interval is continuous and differentiable, then
(1) If the function is satisfied in the interval, then the function is increasing function in the interval and the function curve is concave;
(2) If the function is satisfied in the interval, the function is a decreasing function in the interval and the function curve is concave.
Local extremum
When we say that it reaches the maximum at point, we mean that it is the maximum of the field, as shown in the figure. The maximum value is reached at the point, although = is not the maximum value in the whole image, it is only the maximum value in the point domain, and the other maximum value is B=, which is only the function value at the end of the interval [,], and = is the maximum value of the whole image.
Similarly, reaching the minimum at a point means the minimum in the field, as shown in the figure. The minimum value is reached at the point, although = is not the minimum value in the whole image, it is only the minimum value in the point domain, and the other minimum value is A=, which is only the function value of the function at the end of the interval [,], and = is the minimum value of the whole image.
The concepts of maximum and minimum of a function are partial. If it is the maximum (or minimum) of the function, it is only the maximum (or minimum) of the function in the local range near the point, not necessarily the maximum (or minimum) of the function in the whole definition domain.
We mentioned in the section of differential mean value theorem that if a function is differentiable and the point is its extreme point, then the point must be its stagnation point, but the stagnation point of the function is not necessarily its extreme point. For example, point =0 is the stagnation point of the function, but the internal function is monotonically increasing, so point =0 is not its extreme point, so we can see that the stagnation point of the function is only a possible extreme point. In addition, a function may get an extreme value at its non-derivative point, for example, a function is non-derivative at point =0, but gets a minimum value at that point.
Maximum and minimum value
On the basis of the extreme value discussed above, we further discuss the solution of the maximum and minimum value of the function in the interval. Maximum and minimum values are widely used. People should pay attention to efficiency in everything, from making daily appliances to production, scientific research and various commercial activities, we should consider how to get the maximum output with the minimum input. This kind of problem can often be summed up in mathematics as finding the maximum and minimum value of a function in a certain interval.
Now let the function be continuous in the closed interval and differentiable in the open interval. According to the properties of continuous function in closed interval, the maximum and minimum of function must exist in closed interval. Secondly, if the maximum or minimum value is obtained at a certain point in the open interval, then this maximum or minimum value must be the maximum or minimum value of the function. Therefore, this point must be the stagnation point of the function; Finally, the maximum or minimum value of the function can also be obtained at or. Let's look at the process of finding the maximum or minimum through an example.
Example 5 Find the maximum and minimum values of a function in a closed interval.