To verify that the function $f(x) = x^2 - 4x + 1$ satisfies the three hypotheses of Rolle's Theorem on the interval $[0, 4]$, let's go through each condition and then find the values of $c$ where $f'(c) = 0$.

Step 1: Verify the Hypotheses of Rolle's Theorem

Rolle's Theorem states that for a continuous function $f(x)$ on a closed interval $[a, b]$, if:

1. $f(x)$ is continuous on $[a, b]$,
2. $f(x)$ is differentiable on $(a, b)$,
3. $f(a) = f(b)$,

then there exists at least one value $c$ in $(a, b)$ such that $f'(c) = 0$.

Hypothesis 1: Continuity on $[0, 4]$

Since $f(x) = x^2 - 4x + 1$ is a polynomial, it is continuous on any interval, including $[0, 4]$.

Hypothesis 2: Differentiability on $(0, 4)$

Polynomials are differentiable on any open interval, so $f(x)$ is differentiable on $(0, 4)$.

Hypothesis 3: $f(0) = f(4)$

Calculate $f(0)$ and $f(4)$: $f(0) = 0^2 - 4(0) + 1 = 1$ $f(4) = 4^2 - 4(4) + 1 = 16 - 16 + 1 = 1$ Since $f(0) = f(4) = 1$, the third hypothesis is satisfied.

Step 2: Find $c$ Such That $f'(c) = 0$

Since the function satisfies all hypotheses, there must be at least one value $c$ in $(0, 4)$ such that $f'(c) = 0$.

1. First, find $f'(x)$: $f(x) = x^2 - 4x + 1$ $f'(x) = 2x - 4$

2. Set $f'(x) = 0$ and solve for $x$: $2x - 4 = 0$ $2x = 4$ $x = 2$

So, the value of $c$ where $f'(c) = 0$ is $c = 2$.

Summary

• $f(x)$ is continuous on $[0, 4]$.
• $f(x)$ is differentiable on $(0, 4)$.
• $f(0) = f(4) = 1$.
• By Rolle's Theorem, $c = 2$ is the value in $(0, 4)$ such that $f'(c) = 0$.

If you have further questions or want more details on any step, feel free to ask!

Related Questions

1. How do we know that a polynomial function is continuous on any interval?
2. What are the implications if $f(a) \neq f(b)$ for Rolle's Theorem?
3. Can Rolle's Theorem be applied to intervals where the function has discontinuities?
4. Why is differentiability important in applying Rolle's Theorem?
5. How does Rolle's Theorem relate to the Mean Value Theorem?

Tip:

Remember, Rolle’s Theorem is a special case of the Mean Value Theorem where the endpoints have equal function values!