F (x) = - f (x + 3) find the period of this function Replace it with the form f (x) = f (x + n), where n is the period

F (x) = - f (x + 3) find the period of this function Replace it with the form f (x) = f (x + n), where n is the period

Answer t = 6
Method: 1. Because f (x) = - f (x + 3)
2. Similarly, f (x + 3) = - f (x + 6)
3. Substitution f (x) = - f (x + 3) = - [- f (x + 6)]
=f(x+6)
4.T=6

It is known that a belongs to R, function f (x) = A / x + lnx-1, G (x) = (lnx-1) e ^ x + X (where e is the base of natural logarithm) 1. Find the minimum value of function f (x) on interval (0, e)

(1) The derivative of F (x) is: F '(x) = - A / x ^ 2 + (1 / x)
Let f '(x) > = 0 and get x > = a
f‘（x）

It is known that a belongs to R, the function f (x) = ax LNX, and X belongs to (0, e], (where e is the base of the natural logarithm and is a constant) (1) When a = 1, find out whether there is a real number a between the monotone interval and the extreme value (2) of F (x), so that the minimum value of F (x) is 3. If so, find the value of A. if not, explain the reason

(1) F (x) = x-lnx, f '(x) = 1-1 / x, Let f' (x) = 0, get x = 1, we can know that (0,1) monotonically decreases, (1, e] monotonically increases, and the extreme value f (1) = 1
(2) (0,1) monotonically decreasing, (1, e] monotonically increasing, f '(x) = A-1 / x, f (1 / a) minimum 1-ln1 / a = 3, a = e ^ 2

Known function f (x) = e ^ x + ax, G (x) = (e ^ x) LNX (E is the base of natural logarithm) If f '(x) > 0 is constant for any x ∈ R, try to determine the value range of real number a?

f'(x)=e^x+a,
f''(x)=e^x>0
So f '(x) is an increasing function
f'(x)|min=f'(-∞)=a>0
Therefore, the value range of a is a > 0

High school derivative application Let f (x) = x ^ 3 + ax ^ 2-A ^ 2x + 1, G (x) = ax ^ 2-2x + 1, where the real number a is not equal to 0, if a > 0, 1 when the image of function y = f (x) and y = g (x) has only one common point When G (x) has a minimum value, record the minimum value H (a) of G (x) and find the value range of H (a) 2. If both f (x) and G (x) are increasing functions in the interval (a, a + 2), find the value range of A

Simply put, the process might as well give you ideas
(1) There is a common point that f (x) - G (x) = 0 has and has only one root, because
H (x) = f (x) - G (x) is still a cubic function, so this cubic function has and only has one intersection with the x-axis, which means two possibilities: B ^ 2-4ac of the derivative function H '(x)

1. Find the slope of tangent line at point P by knowing point P (2,8 / 3) on curve y = 1 / 3x ^ (3) From y = 1 / 3x ^ (3), △y=1/3(x+△x)^（3）-1/3x^（3） ① =1/3[3x^（2）△x+3x(△x)^（2)+(△x)^（3)] ② …… Question: How did ① change to ②? 2. The tangent equation of function f (x) = - (1 / x) at point (1 / 2, - 2) is () Question: how to find f '(x) in fractional form? Back to paopaoshi: I know that the slope can be easily obtained by using the derivation formula. I just want to know how the third power is transformed. I can understand that it is difficult to type it with the keyboard. Is there any way to describe it in language? Is there any formula?

I don't know if you don't know: Question 1: first, if you use the derivative formula to get 4, but follow your practice, because you don't know (x + @ x) ^ 3 (sorry, the mobile phone doesn't have a triangle)... In fact, the place turns into (x + @ x) ^ 2 * (x + @ x). Then you even go to... Question 2 has such a formula, which is hidden: