Solution for Exercise 5.3:

1. Let $f(x) = x^2 - x$:

(a) An interval on which $f$ satisfies the hypotheses of Rolle's Theorem:

To satisfy Rolle's Theorem:

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

Choose the interval [0, 1] since:

• $f(x)$ is continuous and differentiable everywhere (as it's a polynomial).
• $f(0) = f(1) = 0$, satisfying $f(a) = f(b)$.

Thus, the interval is:
[0, 1].

(b) Find all values of $c$ that satisfy Rolle's Theorem:

Rolle's Theorem states $f'(c) = 0$ for some $c \in (0, 1)$.

1. Compute $f'(x) = 2x - 1$.
2. Solve $f'(c) = 0$: $2c - 1 = 0 \implies c = \frac{1}{2}.$

Thus, the value of $c$ is:
$c = \frac{1}{2}$.

3. Let $f(x) = x^2 - x$:

(a) Find a point $b$ such that the slope of the secant line through $(0, 0)$ and $(b, f(b))$ is 1:

The slope of the secant line is: $\text{slope} = \frac{f(b) - f(0)}{b - 0} = 1.$

Since $f(0) = 0$, this simplifies to: $\frac{b^2 - b}{b} = 1.$

Simplify: $b - 1 = 1 \implies b = 2.$

Thus, the point is:
$b = 2$.

(b) Find all values of $c$ that satisfy the conclusion of the Mean-Value Theorem (MVT) for $f$ on $[0, b]$, where $b = 2$:

The Mean-Value Theorem states $f'(c) = \frac{f(b) - f(0)}{b - 0}$ for some $c \in (0, b)$.

1. Compute the slope of the secant line: $\frac{f(2) - f(0)}{2 - 0} = \frac{2^2 - 2 - 0}{2} = \frac{2}{2} = 1.$

2. Solve $f'(c) = 1$: $f'(x) = 2x - 1 \implies 2c - 1 = 1 \implies c = 1.$

Thus, the value of $c$ is:
$c = 1$.

Final Answers:

1. (a) Interval: [0, 1]
   (b) Value of $c$: $c = \frac{1}{2}$

2. (a) Point $b$: $b = 2$
   (b) Value of $c$: $c = 1$

Would you like detailed steps or have any other questions?

Here are 5 related questions to consider:

1. What happens if $f(x)$ has discontinuities in the interval?
2. How does Rolle's Theorem relate to the Mean-Value Theorem?
3. Can a cubic function satisfy Rolle's Theorem?
4. What if $f(a) \neq f(b)$ for some interval—does MVT still apply?
5. How can we geometrically interpret the value of $c$ in both theorems?

Tip: Always verify the continuity and differentiability conditions before applying Rolle's or Mean-Value Theorems.