Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers "c" that satisfy the conclusion of Rolle's Theorem.
f(x)=sin4pix , [-1/2,1/2]
Well according to Rolle's Theorem, it has to be continuous on [a.b] and it is and differentiable on (a,b) and it is too. Not so sure about f(a)=f(b).
Then I got the derivative = cos(4pix)4pi
I don't know what to do from there on.. Please help!
To verify the third hypothesis of Rolle's Theorem, you need to check if the function is equal at the endpoints of the interval, in other words, if f(-1/2) = f(1/2).
For f(x) = sin(4πx), let's evaluate f(-1/2) and f(1/2):
f(-1/2) = sin(4π(-1/2)) = sin(-2π) = 0
f(1/2) = sin(4π(1/2)) = sin(2π) = 0
Since f(-1/2) = f(1/2) = 0, the third hypothesis f(a) = f(b) is satisfied.
Now, to find the number(s) "c" that satisfy the conclusion of Rolle's Theorem, you need to find the point(s) in the interval (-1/2, 1/2) where the derivative of the function is equal to zero.
The derivative of f(x) = sin(4πx) is given as f'(x) = cos(4πx)(4π).
To find where f'(x) = 0, set cos(4πx)(4π) = 0 and solve for x:
cos(4πx)(4π) = 0
cos(4πx) = 0
To find the values of x, we know that the cosine function equals zero at every odd multiple of π/2. So solve:
4πx = π/2, 3π/2, 5π/2, ...
Simplifying:
x = 1/8, 3/8, 5/8, ...
These values are in the interval (-1/2, 1/2) and satisfy the conclusion of Rolle's Theorem.
Therefore, all the numbers "c" that satisfy the conclusion of Rolle's Theorem are c = 1/8, 3/8, 5/8, ... within the interval (-1/2, 1/2).