Mathematics elective course for senior two 2-1 If point P is any point on hyperbola x ^ 2 / 16-y ^ 2 / 9 = 1 which is different from the vertex, and F1 and F2 are two focal points, what is the trajectory equation of the center of gravity of triangle pf1f2?

Mathematics elective course for senior two 2-1 If point P is any point on hyperbola x ^ 2 / 16-y ^ 2 / 9 = 1 which is different from the vertex, and F1 and F2 are two focal points, what is the trajectory equation of the center of gravity of triangle pf1f2?

a²=16,b²=9
c²=25
c=5
F1(-5,0),F2(5,0)
P(m,n)
Center of gravity m (x, y)
Then x = (- 5 + 5 + m) / 3, y = (0 + 0 + n) / 3
m=3x,n=3y
P on hyperbola
m²/16-n²/9=1
Substituting
9x²/16-y²=1
N is not equal to 0
So y is not equal to 0
therefore
9x & sup2 / / 16-y & sup2; = 1, remove (± 4 / 3,0)

F (x) = (the square of x-3x + 1) e, and find the points on the function curve, so that the tangent of the curve at these points is parallel to the X axis

Because f (x) = (x ^ 2-3x + 1) e ^ x, F "(x) = (x ^ 2-x-2) e ^ X
Because e ^ x > 0 is always true, only x ^ 2-x-2 = 0 is needed to satisfy the condition, so x = - 1 or 2
So at point (- 1,5 / E) or point (2, - e ^ 2)

1: prove that when x is greater than 0, SiNx is less than x

Certification:
Constructor f (x) = sinx-x
F '(x) = cosx-1 ≤ 0
So f (x) decreases on (0, + ∞)
So f (x)

At present, there are 2 iPhone 4S and 5 iPhone 3 of the same Apple mobile phone. Four of them are awarded to four excellent employees of the company, one for each. How many different reward methods are there? The answer is 10. Why are they all in the same order,

C42+C41=10
The first step is to choose four mobile phones, and the second step is to give them to others
There are only two results when choosing a mobile phone:
One is that there is a 4S, three 5S, then C41, that is, one person got the iPhone 5
Second, there are two 4S and two 5, then C42, which is to get the iPhone 5 from two of four people