1 -- the multiple of 3 in 100

3.6.9.12.15.18.21.24.27.30.33.36.39.42.45.48.51.54.57.60.63.66.69.72.75.78.81.84.87.90.93.96.99
33 in total

In 1 ~ 100, which are the multiples of 2 and 3?
Tell them all,

6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96,12 are multiples of 6

What are the multiples of 3
Within 100 million

Are there any restrictions?
Some words: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42. There are countless~
If there is no limit: let me teach you a formula - the sum of all digits is a multiple of three, then the number must be a multiple of three!
Example: if 121 + 2 = 3 / 3 = 1 is a multiple of three, then 12 / 3 = 4 must be a multiple of three
The landlord chooses me!

Solving monotone interval problem with derivative
The solution set of the inequality f '(x) > 0 is the monotone increasing interval of y = f (x), and the inequality f' (x) is the monotone increasing interval of y = f (x)

F '(x) > 0 is a sufficient but not necessary condition for f (x) monotonically increasing,
That is: from F '(x) > 0, we can deduce f (x) monotone increasing, but from F (x) monotone increasing, we can not deduce f' (x) > 0. (for example, function f (x) = x & # 179;)
F '(x) > = 0 is a necessary but not sufficient condition for f (x) monotonically increasing,
That is: from F '(x) > = 0, we can not deduce the monotone increasing of F (x) (for example, the function f (x) = 4), but from F (x) monotone increasing, we can deduce f' (x) > = 0
Therefore, when a function is known to be monotone in a certain interval, the value range of a parameter is usually given an equal sign, while when a monotone interval is given, it is usually not given an equal sign

How to prove function monotony with derivative
For example, y = x ^ 3
How to prove monotony with derivative?

Y '= 3x ^ 2 > = 0 and y = 0 only when x = 0,
So y = x ^ 3 is monotonically increasing on R

How to prove monotonicity of check function with derivative
I'm a freshman in high school. It's too boring to prove the difference method for monotonicity of check function. The teacher said to do it with derivative. The detailed process and method of using derivative to prove monotonicity of check function. OK, add 30 points! Never break your promise!
For example, f (x) = x + 1 / x, write down the proof process, and then summarize the method,

Given that f (x) = x + 1 / x, we can get f '(x) = 1-1 / x ^ 2 = (x ^ 2-1) / x ^ 2 by derivation. Then let f' (x) = 0, we can get x = 1 or x = - 1
Proper list x

In mathematics books, it is said that the derivative is greater than 0, and the function increases monotonously. I think, in any case, first the derivative is greater than or equal to 0, and then discuss whether it is true that the derivative is equal to 0
It is said in the mathematics book that the derivative is greater than 0 and the function increases monotonically. I think that no matter what the case is, first the derivative is greater than or equal to 0, and then exclude the case that the derivative is in a section or constant to 0 (when the original function is parallel to the X axis, it is not tenable). Therefore, I think what is said in the book is not accurate,

"Derivative greater than 0, monotone increasing function" is undoubtedly a true proposition,
You are also right in this case, but in some cases, if the derivative is greater than or equal to 0, the function will increase monotonously, but in some cases, it can not be ruled out that the function is always 0
In order to avoid this misunderstanding, the textbook only lists the case greater than 0

On the question of the definition of derivative, if f '(x) is greater than 0 on R, then can we deduce that f (x) is a monotone increasing function on R?
My question is, if f (x) is a piecewise function, it will increase on each segment, but not on R as a whole

Your problem is that the original definition of derivative is not fully understood,
For piecewise function, there is no derivative at the piecewise point, so we only discuss the continuous interval
If a function has a derivative on an interval, it must be a continuous interval

If the function increases monotonically in (a, b), can the derivative be zero in (a, b)?

It is possible, for example, that f (x) = x ^ 3 increases monotonically on (- 1,1), but f '(0) = 0

Derivative extremum problem, after the list of monotone increasing function monotone decreasing function how to judge
List, bring in a value on the interval, less than 0
The function on the interval is a decreasing function, and the original function on the interval greater than 0 is an increasing function?

The interval whose derivative is less than 0 is a decreasing function
The interval whose derivative is greater than 0 is an increasing function
In addition: Generally speaking, the value of the independent variable whose derivative is equal to 0 is the extreme point
To solve the problem, we should substitute a value in the interval into the derivative function

1 -- the multiple of 3 in 100