How to tell that a quadratic equation is an increasing function in an interval

Suppose a quadratic equation.

ax 2+bx+c=0 where a<>0, where it is determined whether it is an increasing function in the (m,n) interval (n>m).

Methods to do this:

1。Suppose in the interval (m,n), y=ax 2+bx+c In this way, the equation is converted into the form of a function.

If the original equation has two solutions x1 and x2, i.e. the two intersections of the function y=ax 2+bx+c and the x-axis, then y=ax 2+bx+c is the axis of symmetry.

is x=(x1+x2) 2, in this case, the function is determined by the positive or negative of a, and which part of the symmetry axis (m,n) is in the interval to determine whether the function is an increasing function in the interval.

2。If there is only one solution of the original equation x1=x2, then the increase or decrease of the function can be judged directly according to the magnitude of a, judging that (m,n) is around x1.

3。A situation where there is no solution to the equation. Find the axis of symmetry x=-b (2a) and determine which direction (m,n) is in the axis of symmetry.

To sum up, in fact, it is. Convert the equation into a functional form, according to a quadratic function.

characteristics to judge.

A is greater than 0, the opening is upward, and the left side of the axis of symmetry is the subtractive function.

On the right is the increment function, and y has the minimum point.

A is less than 0, the opening is downward, the left side of the axis of symmetry is the increase function, the right side is the subtraction function, and y has the maximum value point.

At this time, there are generally only three situations that may occur in the interval:

Increasing functions, subtracting functions, or partially subtracting functions, partially subtracting functions.

The derivative method mentioned upstairs is very simple, but it is not learned until the second year of high school.

If the opening is up, the right side of the axis of symmetry is the increment function.

If the opening is downward, to the left of the axis of symmetry is the increment function.

If the result is positive, the original function is an increasing function, and if it is negative, it is a subtraction function. If it is 0, the original function is a straight line.

() From the meaning of the question, the discriminant formula can be obtained, and the scope of the requirements of this solution. () is divided into two cases, at that time and at that time, respectively, using the grade inequality, to obtain the range. Solution:

From the meaning of the question can be obtained discriminantly, the solution can be obtained, or, so the scope of the requirement is, or. () is known from (), then, if and only then, take the equal sign, that is, at that time, obtain the minimum value as. Therefore, there is.

At that time, if and only if, instantly, take the equal sign, and in summary, the range of the function is . This question mainly examines the application of finding the maximum value of quadratic functions in closed intervals and the properties of quadratic functions, which embodies the mathematical ideas of classification and discussion, and belongs to the basic questions.

Analysis: The value range (1) of f(x)=ax +bx+c(a≠0) on [m,n] is found f(m), f(n), f(-b 2a)(2)fmin=min{f(m),f(n),f(-b 2a))fmax=max{f(m),f(n),f(-b 2a))(3) respectively.

fmin，fmax]