Question

How do you verify that the function `f(x)= sin(22pix)` satisfies the three hypotheses of Rolle's Theorem on the given interval [-1/11, 1/11] and then find all numbers c that satisfy the conclusion of Rolle's Theorem?

Answer 1

In this context, "Verify" mean "Prove" or "Show". Think of this as a writing assignment.

"Convince a reader that the function `f(x)= sin(22pix)` satisfies the three hypotheses of Rolle's Theorem on the interval `[-1/11, 1/11]` "

(Then do some algebra to prove to your grader (1) I know what the conclusion says and (2) I can do algebra to solve an equation.)

Go through the hypotheses in order and tell you reader why they should agree that the hypothesis is true for this function on this interval:

H1: The function `f(x)= sin(22pix)` is continuous on the closed interval `[-1/11, 1/11]` , because:

the 'inside' function `22 pi x` is linear (polynomial) and linear functions are continuous at all real `x` and the 'outside' function, `sinx` is continuous for all real `x` and, finally, the composition of continuous functions is continuous.

H2: The function `f(x)= sin(22pix)` is differentiable on the open interval `(-1/11, 1/11)` , because:

`f'(x) = 22 pi cos(22 pi x)` exists for all `x` in the interval (in fact, it exists for all real `x`, but here we only need this little bit of it)

H3: The function has equal values at the endpoints or the interval

`f(-1/11)= sin(22pi (-1/22)) = sin(-2 pi) = 0` and

`f(1/11)= sin(22pi (1/22)) = sin(2 pi) = 0`

We have now verified that the function `f(x)= sin(22pix)` satisfies the three hypotheses of Rolle's Theorem on the interval `[-1/11, 1/11]`

The conclusion of Rolle's Theorem tells us that there is a number `c` in the open interval `(-1/11, 1/11)` such that `f'(c)=0`

In this case, there is a number `c` in the open interval `(-1/11, 1/11)` such that `22 pi cos(22 pi c) = 0`

From trigonometry we know that the solutions are:

`22 pi c = pi/2 + 2k pi` for integer `k`.

So `22 c = 1/2 +2k` and `c = 1/44 +k/11`

Taking `k=0` or `-1` will give `c` in `(-1/11, 1/11)`