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 that the function satisfies the three hypotheses of Rolle's Theorem, let's go through each hypothesis one by one.
1. Continuity: The function f(x) = sin(4πx) is continuous on the closed interval [-1/2, 1/2] because it is a trigonometric function of x which is continuous for all real numbers. So, the first hypothesis is satisfied.
2. Differentiability: The function f(x) = sin(4πx) is differentiable on the open interval (-1/2, 1/2) because it is the composition of the differentiable function sin(x) and the linear function 4πx. Both sin(x) and 4πx are differentiable functions. So, the second hypothesis is satisfied.
3. Equality of Function Values: Now we need to check if f(-1/2) = f(1/2).
Evaluating f(-1/2):
f(-1/2) = sin(4π*(-1/2)) = sin(-2π) = 0.
Evaluating f(1/2):
f(1/2) = sin(4π*(1/2)) = sin(2π) = 0.
Since f(-1/2) = f(1/2) = 0, the third hypothesis is also satisfied.
Now, let's find all the numbers c that satisfy the conclusion of Rolle's Theorem, which states that there must exist at least one number c in the open interval (-1/2, 1/2) such that f'(c) = 0.
The derivative of f(x) = sin(4πx) is f'(x) = cos(4πx)*4π.
To find the values of c that satisfy f'(c) = 0, set f'(x) equal to zero and solve for x:
cos(4πx)*4π = 0.
Since cos(4πx) cannot be equal to zero for any real value of x, the only solution to the above equation is:
4πx = 0.
Dividing both sides by 4π, we get:
x = 0.
Therefore, the conclusion of Rolle's Theorem is satisfied for x = 0.
In summary, the function f(x) = sin(4πx) satisfies the three hypotheses of Rolle's Theorem on the interval [-1/2, 1/2], and the only number c that satisfies the conclusion of the theorem is c = 0.