Answer:
Step-by-step explanation:
Squaring both sides of the equation would give which then simplifies to
We can then subtract 7 from both sides to get
Then we can subtract x from both sides, which cancel out and we get
Then we need to divide both sides by -2 to get
Then we need to square both sides to get rid of the square root to get 9 = x + 6
Then finally subtracting 6 from both sides we get 3 = x
Explanation:
In order to prove that affirmation, we define the function g over the interval [0, 1/2] with the formula
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
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
Hence, f(c+1/2) = f(c) as we wanted.