Real Analysis: Differentiation and Mean Value Theorem

Prove that if a function is differentiable at some point $C$, then it is also continuous at that point $C$.

Prove Rolle's Theorem: Suppose $f$ is continuous on a closed interval $[a, b]$ and differentiable on the corresponding open interval $(a, b)$. Show that if $f(a) = f(b)$, then there exists a $c$ in $(a, b)$ such that $f'(c) = 0$.

Prove the Mean Value Theorem: If $f$ is continuous on a closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists a $c$ between $a$ and $b$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Prove the Mean Value Theorem.

Find the critical point, any local maximums or minimum, and the intervals where the function is increasing and decreasing for $y = 2x^2 - 8x + 6$.

Find the intervals where the function $y = 2x^3 + 3x^2 - 12x + 8$ is increasing and decreasing, and find any maximum or minimums, and find the critical points.

Find the critical numbers for a given function $f(x)$, and determine whether any relative extrema occur at those points.

Find the value of $c$ for the function $f(x) = x^2 - 2x + 1$ on the interval $[-1, 3]$ where $f'(c) = 0$ using Rolle's Theorem.

Evaluate if Rolle's Theorem applies for the function $f(x) = \frac{x^2 - 4}{x}$ on the interval $[-2, 2]$.

Verify Rolle's Theorem and find the value or values of c that satisfy it for the following function on the given interval.