Function equation f (x) - 2F (- x) = x ^ 2-5x, find f (x) The solution given by the answer is to construct another equation f (- x) + 2F (x) = x ^ 2 + 5x, but I don't know the basis of such construction?

Function equation f (x) - 2F (- x) = x ^ 2-5x, find f (x) The solution given by the answer is to construct another equation f (- x) + 2F (x) = x ^ 2 + 5x, but I don't know the basis of such construction?

Replace X in the formula with - x
f(-x)-2f(x)=x^2+5x
The combination of Shangshi and Yuanshi
We get f (x) = - x ^ 2 - (5 / 3) X

The linear function f (x) satisfies f (x + 1) - 2F (x-1) = 5x + 1

Let f (x) = ax + B
Then f (x + 1) - 2F (x-1) = a (x + 1) + B-2 (a (x-1) + b)
=-AX+3A-B=5X+1
Equality identity
Necessary - a = 5 3A - B = 1
==>A=-5,B=-16
So f (x) = - 5x-16

Find the maximum value of the function f (x) = min {(3 / 2x) + 3, (- 7 / 5x) + 7, (1 / 4x) + 2}

f(x)=min{(3/2x)+3,(-7/5x)+7,(1/4x)+2}
It is the one with the smallest median value of the three lines, and the equations are listed respectively to find out the points where the lines intersect each other
Then find out the minimum value in the interval, and the maximum value that can be obtained is actually the intersection value

If the function whose domain is r satisfies f (x) + F (x + 2) = 2x & sup2; - 4x + 2, f (x + 1) - f (x-1) = 4 (X-2), find f (x)

F (x + 1) - f (x-1) = 4 (X-2), substituting x + 1 = X
f(x+2)-f(x)=4(x+1-2)=4(x-1)
Subtracting two equations
f(x)=x²-2x+1-2(x-1)=x²-4x+3