Explanation:
In order to prove that affirmation, we define the function g over the interval [0, 1/2] with the formula ![g(x) = f(x+1/2)-f(x) .](https://tex.z-dn.net/?f=%20g%28x%29%20%3D%20f%28x%2B1%2F2%29-f%28x%29%20.%20)
If we evaluate g at the endpoints we have
g(0) = f(1/2)-f(0) = f(1/2) - f(1) (because f(0) = f(1))
g(1/2) = f(1) - f(1/2) = -g(0)
Since g(1/2) = -g(0), we have one chance out of three
- g(0) > 0 and g(1/2) < 0
- g(0) < 0 and g(1/2) > 0
- g(0) = g(1/2) = 0
We will prove that g has a zero on [0,1/2]. If g(0) = 0, then it is trivial. If g(0) ≠ 0, then we are in one of the first two cases, and therefore g(0) * g(1/2) < 0. Since f is continuous, so is g. Bolzano's Theorem assures that there exists c in (0,1/2) such that g(c) = 0. This proves that g has at least one zero on [0,1/2].
Let c be a 0 of g, then we have
![0 = g(c) = f(c+1/2)-f(c)](https://tex.z-dn.net/?f=%200%20%3D%20g%28c%29%20%3D%20f%28c%2B1%2F2%29-f%28c%29%20)
Hence, f(c+1/2) = f(c) as we wanted.
For the answer to the question above
asking how much water (W) is used when there are no (n) marbles in the tank? W(0)=32-0.05*(0)
<u><em>W(0) = 32 liters</em></u>
if Andrei uses 150 marbles?
W(150)=32-0.05*(150) <u><em>
</em></u><u><em> W(150 )= 24.5 liters</em></u>I
hope this helps. Have a nice day!
2ab - 3a = a.(2b-3)
now the same thing
let's say n=1
a.(2b-3) + n.(2b-3) = (2b-3).(a+n)
Understood?
Answer:
3(14 + x) = 57
Step-by-step explanation:
I just answered the same question edmentum