Image crossing point P (0,1) of even function f (x) = ax ^ 4 + BX ^ 3 + CX ^ 2 + DX + e And the tangent equation at x = 1 is y = X-2 to find the analytical formula of F (x) Derivative method

Image crossing point P (0,1) of even function f (x) = ax ^ 4 + BX ^ 3 + CX ^ 2 + DX + e And the tangent equation at x = 1 is y = X-2 to find the analytical formula of F (x) Derivative method

Very simple. First, we know that f (0) = 1, deduce e = 1, and the tangent equation at x = 1 is y = X-2  , It can be seen that the even function passes through the point (1, - 1), and the derivative of F (x) = ax ^ 4 + BX ^ 3 + CX ^ 2 + DX + 1 is f (x) = 4ax ^ 3 + 3bx ^ 2 + 2cx + D, so f '(1) = 1. Because it is an even function, it passes through the point (- 1, - 1) and its derivative on (- 1, - 1), so four equations about ABCD can be obtained: a = 5 / 2, B = 0, C = - 9 / 2, D = 0. Therefore, the analytical formula of F (x) is y = (5 / 2) x ^ 4 - (9 / 2) x ^ 2 + 1
The image is roughly

It is known that (x + 2) = the fifth power of AX + the fourth power of BX + Cx ³＋ dx ²＋ Ex + F find 16b + 4D + F Yes (x + 2) On the left is (x + 2)

On the left is x + 2
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According to the question, let x = - 2. The solution is:
-32a+16b-8c+4d-2e+f=0
Let x = 2 solve:
32a+16b+8c+4d+2e+f=(2+2)=4
Add the following formula to the above formula:
32b+8d+2f=4
So: 16b + 4D + F = 22

The fourth power of AX + BX ³+ cx ²+ DX + e = (X-2) to the fourth power, find the value of a + B + C + D + E

because
ax^4+bx ³+ cx ²+ dx+e=(x-2)^4
Let x = 1
So a + B + C + D + e = (1-2) ^ 4 = 1

1. The fourth power of known ax + BX ³+ cx ²+ DX + e = (X-2) to the fourth power = (X-2) to the fourth power (1) Find the value of a + B + C + D + E (2) Find the value of a + C 2. If polynomial (2mx) ²- y ²+ 3x+1）-（5x ²- 5y ²+ The value of 3x) is independent of X, find 4m ²- (4m-5) + 6m 3. China's taxi charging standards vary from place to place. In city a, the starting price is 6 yuan and 1.5 yuan per kilometer after 3km; City B: the starting price is 10 yuan, and 1.2 yuan per kilometer after 3km? (1) How much is the difference in payment for taking a taxi s (s > 3) km in cities a and B? (2) If the distance by taxi in cities a and B is 10km, which city has a higher charge? How much higher?

1. Solution 1). If x = 1, then a + B + C + D + e = (1-2) ^ 4 = 12. If x = - 1, then A-B + C-D + e = (- 1-2) ^ 4 = 81, then a + C + e = 41, and then x = 0, that is, e = (- 2) ^ 4 = 16, so a + C = 252. Solution. The value of the original formula = (2m-5) x ^ 2 + 4Y ^ 2 + 1 has nothing to do with X, then 2m-5 = 0, that is, M = 5 / 2. Then 4m ^ 2 - (4m-5) + 6m = 25-5 + 15 = 353. Solution. 1)

Let f (x) be continuous and differentiable on [0,1], and f (0) = 0,0 ≤ f '(x) ≤ 1. Test: [∫ ^ (0,1) f (x) DX] ^ 2 ≥ ∫ ^ (0,1) [f (x)] ^ 3DX Although I want to start with 0 ≤ f '(x) ≤ 1 to explain f (x) > [f (x)] ^ 3, it seems useless

Simultaneous derivation on both sides
Or subtraction

It is known that f (x) has a second-order continuous derivative, and f (0) = 1, f (2) = 4, f '(2) = 2. Find ∫ XF' '(2x) DX

∫(0→1) xƒ''(2x) dx= (1/2)∫(0→1) xƒ''(2x) d(2x)= (1/2)∫(0→1) x d[ƒ'(2x)]= (1/2)[xƒ'(2x)] |(0→1) - (1/2)∫(0→1) ƒ'(2x) dx= (1/2)xƒ'(2x) - (1/4)ƒ(2x) |(0→1)= [...