High school proves monotonicity of exponential function y = 2 x proves monotonicity. I'm in grade one. Can you use a simpler definition, such as monotonicity? In addition, the situation I encountered in the certificate was also mentioned. The following is the wrong solution:
Solution 1: let x 1 < x2, let c = x2-x 1 > 0.
f(x 1)-f(x2)=2^x 1-2^x2=2^x 1( 1-x^c)
∫c > 0
1< x c (how did you get here? Isn't it proved by the definition of monotonicity? )
Solution 2: let x 1 < x2, and let c = x2-x 1 > 0.
F(x 1) divided by f (x2) = 2 (x 1-x2).
∫x 1-x2 < 0
∴ 2 (x 1-x2) < 2 0 = 1 (isn't this also using monotonicity to prove monotonicity? )
To find the positive solution of monotonicity definition, there is no problem of circular argument. In the two proofs, we use the property that the positive power of 2 is greater than 1, which is not the inference of monotonicity of exponential function, but can be directly deduced from the definition of exponent. The problem is that we can't explain how to define the quadratic root of 2 in high school, so we can't prove this property directly. Because the power of rational numbers is defined.
The positive integer powers of 1 and 2 are greater than 1. This can be proved by induction. n = 1，2 >； 1,n=k,2^k>； 1,n=k+ 1,2^n=2^(k+ 1)>； 2> 1, so this proposition is applicable to positive integers.
2. The positive integer power of a positive number less than 1 is less than 1. This can also be proved by induction.
The power of the positive rational numbers of 3 and 2 is greater than 1. This can be proved by reducing to absurdity. The power of the positive rational number of (1)2 is greater than 0. It seems obvious, but it still needs to be proved. (2) If the power of the positive rational number of 2 is less than 1, it is less than 1. A rational number is a positive number multiplied by an appropriate number. So the power of this number must be a positive integer power of 2, so that the positive integer power of 2 will be less than 1. This is in contradiction with the point 1. So we can know that the positive rational power of 2 is greater than 1. The proposition is extended to irrational numbers, which is not what I can say.
Obviously, neither of the two methods you gave to prove monotonicity has the problem of circular argument.