Derivative of mathematics in the third year of high school, derivative of mathematics in the third y

Consider a>1 and 1>a>0.

1) When 1>a>0.

Pictorial considerations. a x decreases as x increases, the function approaches 0x when x approaches infinity, decreases as x increases, and the function approaches negative infinity when x approaches infinity.

So in this case, f(x)=a x-x cannot be evergreen at 02)a>1.

The value of the function decreases first and then increases.

Functions are continuously derivable within a defined domain (the entire real space).

f'(x)=lna*a^x-1

f'(x)=0 x=loga(1 lna) verify: x=loga(1 lna) f''(x) is not equal to 0, so it is an extreme point (minimum).

In this case, f(x0)=(1 lna) -loga(1 lna)=[1-ln(1 lna)] lna

When a>e (1 e), there is f(x0)>0———i.e., the minimum point value of 0, and the function is evergreen in the defined domain to 0).

So a>e (1 e) is the solution to the problem.

I don't know if there is a mistake, the idea is like this, I hope it can help you

It's really not right.

f'(a)=0 means that f(x) has a tangent slope of 0 at a, but a point with a tangent slope of 0 is not necessarily an extreme point.

For example, f(x)=x 3, where x=0 is f'(x)=0, but the original function has no extremum at x=0.

If you don't understand, please ask.

f'(x)=a^x lna-1

Believe me, I'm from the university's mathematics department.

Hello this student, this question is still relatively difficult, especially the second question, I hope it will help you, you can like it, I wish you a happy study.

f'(x)=-1/(2-x)+2ax

Tangent slope at point (1, f(1)).

f'(1)=-1 (2-1)+2a=2a-1 and f(1)=a

Then the equation for the straight line is:

y-a=(2a-1)(x-1)

l and the garden (x+1) 2+y 2=1, then.

The distance from the center of the circle to the straight line is greater than the radius.

2(2a-1）-a|[(2a-1) 2+1]>1 |3a-2|/√[(2a-1)^2+1]>15a^2-8a+2>0

then (4-6) 50

Then: 0f'(x)=-1/(2-x)+2ax=0

Then: 2ax 2-4ax+1=0

x=a+√(a^2-a/2),x=a-√(a^2-a/2),f''(x)=-1/(2-x)^2+2a

With the entry point value, then.

f''(a- (a 2-a 2))) <0 to achieve the maximum value...

The answer is to talk about Lao-LNA-1, and the specific calculation of the elevation containing reeds is shown in the following diagram.

Derivative of the function.

y'=a 2e (ax) * x (x-2 a) when x>0 or x<-2 a, y'>0, then the function is an increasing function;

When -2 a

I've returned to the teacher, it's too hard.