Can anyone give a rigorous proof of the monotonicity of exponential functions?

For a x, a > 0, it is impossible to discuss its monotonicity without first stating its exact definition.

The exponential function is defined over the entire range of real numbers. Let's start with the definition of an integer:

a n = a * a * a (n > 0, the same below) (n a multiplication).

a^0 = 1

a^(-n) = 1 / a^n

Let's talk about the definition on the set of rational numbers:

a(1 n) = n arithmetic roots of a, a(p q) = q arithmetic roots of (a p), where p q is the reduced fraction.

In this way, the exponential function on the set of rational numbers is defined. And it is not difficult to prove the monotonicity of a (p q) on a set of rational numbers by elementary methods. In fact, for A (P1 Q1) and A (P2 Q2), the fractions P1 Q1 and P2 Q2 can be divided together, so that the denominator is the same, and the denominator is the same, so that they are P1 respectively' / q， p2' / q。

Now it's a (1 q) base and p1'and p2'is the size of the two numbers of the exponent. Obviously, when a > 1, a (1 q) >1, so that the function is strictly monotonically increased; Conversely, a < 1, it can also be proved that the function is strictly monotonically reduced.

Now for any real number x, we can take a series of rational numbers, and when n increases infinitely, it tends to x monotonically, then we can define a x as the limit of a qn when n increases infinitely.

If we use a series of rational numbers to approximate the exponential function, then for two real numbers that are arbitrarily close to each other, we can take two rational numbers that are accurate enough to replace them, and keep the magnitude relationship constant. (Of course, there are some operational guarantees in this, for example, why the previous definition is reasonable, i.e., why limits exist, etc., I will not write about it.) In this way, the monotonicity of the exponential function under real numbers can be reduced to the monotonicity of rational numbers, and the same as it.

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Ask your math teacher!

The teachers spoke in detail.

First of all, y a x is an exponential function, and we generally discuss the case of a>0 and a≠1.

When the exponent is a negative integer, let =-k, then, obviously, x≠0, the domain of the function is ( 0) (0,+ so we can see that x is limited by two points:

One is that it is possible to be the denominator instead of 0.

One is that it is possible to be under an even number of times and not be negative, then we can know: when less than 0, x is not equal to 0; When the denominator is even, x is not less than 0; When the denominator of is odd, x takes r.

Monotonic interval: When is an integer, the positive, negative and parity determine the monotonicity of the function.

When is a positive odd number, the image is monotonically incremented within the defined domain r.

When is a positive and even number, the image decreases monotonically within the second quadrant and increases monotonically within the first quadrant in the defined domain.

When is a negative odd number, the image decreases monotonically in each quadrant of the first three quadrants (but cannot be said to be monotonically decreasing within the defined domain r).

When negatively even, the image increases monotonically in the second quadrant and decreases monotonically in the first quadrant.

When is a fraction (and the numerator is 1), the positive and negative and the parity of the denominator determine the monotonicity of the function.

How to prove function monotonicity.

There are two main methods for determining the monotonicity of a function over an interval:

Definitions:1Let any x1 and x2 given intervals, and x12Calculate f(x1)- f(x2) to the simplest. [best expressed as the form of integer product].

3.Determine the sign of the above difference.

Derivative method: use the derivative formula to find the derivative, and then judge the magnitude relationship between the derivative function and 0, so as to judge the increase or decrease, the derivative value is greater than 0, indicating that it is a strict increasing function, and the derivative value is less than 0, indicating that it is a strictly decreasing function, provided that the original function must be continuous. When the derivative is greater than or equal to 0, it can also be an increasing function, and the same is true when the derivative is less than or equal to 0.

Extended information: Some functions are monotonic throughout the defined domain; Some functions are increasing over some intervals within the defined domain and subtracting on partial intervals; Some functions are non-monotonic, such as constant functions.

The monotonicity of a function is the "holistic" property of a function in a monotonic interval, which is arbitrary and cannot be replaced by special values.

When using derivatives to discuss the monotonic interval of a function, we should first determine the domain of the function, and in the process of solving the problem, we can only judge the monotonic interval of the function by discussing the symbols of the derivatives in the defined domain.

If there is more than one monotonic interval for a function with the same monotonicity, then these monotonic intervals cannot be concatenated with " ", but only separated by the word "comma" or "and".

Evidence: Calling the absence of a hidden set f(x)=a's x-times, a-0, x-rf'(x)=a's x-power*lna

If a 1, then lna Zheng 0, at which point F'(X) 0, the exponential function increases monotonically.

If a 1, then LNA 0, at this time f'(x) 0, the exponential function is decreasing. Certification.

y=a x If a>1, then the function increases monotonically, if 00y2=5 x2>0

y1/y2=5^x1/5^x2

5^(x1-x2)

Because x1 x2 so x1-x2 0 5 (x1-x2) 1 so y1 y2

According to the definition of the increase function.

y=5 superscript to the power of x, which is an increment function in the defined domain.

The exponential function proves monotonicity with definitions, generally as a quotient, and then compares the magnitude with 1.

Exponential functions are important functions in mathematics. The function applied to the value e is written as exp(x). It can also be written equivalently as ex, where e is a mathematical constant, which is the base of the natural logarithm, approximately equal to , also known as Euler's number.

It occupies a certain place in high school early Ming math. So what are the properties of exponential functions? How to prove the monotonicity of an exponential function?

Exponential functions generally have the following properties:

1) The definition domain of the exponential function is the set of all real numbers, the premise here is that a is greater than 0 and not equal to 1, for a is not greater than 0, then there must be no continuous interval in the definition domain of the function, so we do not consider it, and the meaninglessness of the function that a is equal to 0 is generally not considered.

2) The exponential function has a range of real numbers greater than 0.

3) The function graphs are all concave.

4) If a is greater than 1, the exponential function increases monotonically; If a is less than 1 and greater than 0, it is monotonically decreasing.

5) It can be seen that when a moves from 0 to infinity (which of course cannot be equal to 0), the curve of the function moves from the position of the monotonically decreasing function close to the positive semi-axis of the y-axis and the x-axis, respectively, to the position of the monotonically increasing function of the positive half-axis of the y-axis and the negative half-axis of the x-axis, respectively. where the horizontal line y=1 is a transition position from decreasing to increasing.

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6) Functions always tend infinitely towards the x-axis in one direction and never intersect.

7) The function always passes through the point (0,1), (if y=ax+b, then the function is fixed through the point (0,1+b) (8) Obviously, the exponential function is unbounded.

9) Exponential functions are neither odd nor even.

10) When a in two exponential functions is reciprocal to each other, the two functions are symmetric with respect to the y-axis, but neither function is parity.

The monotonicity of a direct function can be easily demonstrated by understanding the following points:

1) The exponential function is defined in the domain r, where the premise is that a is greater than 0 and not equal to 1. If a is not greater than 0, it will inevitably make the definition domain of the function discontinuous, so we do not consider it, and the function that a is equal to 0 is meaningless is generally not considered.

2) The exponential function has a range of (0, +).

3) The function graphs are all concave.

4) a>1, the exponential function increases monotonically; If.

5) y=ax+b, then the function is fixed over the point (0,1+b))(8) The exponential function is unbounded. (9) The exponential function is a non-odd and non-even function (10) The exponential function has an inverse function, and its inverse function is a logarithmic function, which is a multi-valued function.

The theoretical basis for the monotonicity proof of exponential function is Wang Xiaojun, Bazhong No. 5 Middle School.

Do you have to do it by definition? Or can it be reduced to a function with known monotonicity?

First, define the domain:

The denominator cannot be 0, so 3 x-1≠0,3 x≠1,x≠0

So the domain of f(x) is divided into two separate intervals (- 0) and (0,+ which are discussed separately in these two intervals.

f（x）=（3^x+1）/（3^x-1）

1+（2/（3^x-1））

When x (-0).

3 x-1 0 and monotonically increasing.

So 2 (3 x-1) 0 and monotonically decreasing.

So f(x)=1+(2 (3 x-1)) 1, and decreasing monotonically.

When x (-0).

3 x-1 0 and monotonically increasing.

So 2 (3 x-1) 0 and monotonically decreasing.

So f(x)=1+(2 (3 x-1)) 1, and decreasing monotonically.

So f(x) is a monotonically decreasing function in the two open intervals (- 0) and (0,+).

Note: f(x) is only a monotonically decreasing function in the two open intervals (- 0) and (0,+). , but in the whole defined domain, it is not a monotonically decreasing function, because if you take x1 0, x2 0, it is obvious that x1 x2

Then f(x1) 1 f(x2).

So in the whole defined domain, it is not a monotonically decreasing function, which is why you can't prove its monotonicity with the definition method in the whole definition domain. In addition, 3 x represents 3 to the x power, and the computer cannot put x into the superscript.

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