Rolle's Theorem is a key concept in calculus, bridging continuity and differentiability. It states that for a continuous, differentiable function with equal endpoint values, there's a point where the derivative is zero.

This theorem has practical applications in finding maximum and minimum points. It's crucial for understanding the Mean Value Theorem and forms the foundation for many important calculus proofs and problem-solving techniques.

Rolle's Theorem and Its Applications

Conditions for Rolle's theorem

• Function $f$ must be continuous on closed interval $[a, b]$ meaning there are no gaps or breaks in the graph
• Function $f$ must be differentiable on open interval $(a, b)$ meaning it has a well-defined derivative at every point within the interval (polynomial, rational, exponential, logarithmic, trigonometric functions)
• Function $f$ must satisfy condition $f(a) = f(b)$ meaning the y-values at the endpoints of the interval are equal (graph starts and ends at same height)

Proof of Rolle's theorem

• Extreme Value Theorem states a continuous function $f$ on closed interval $[a, b]$ attains its max and min values, either at an interior point or endpoint (top of hill or bottom of valley within interval or at start/end)
  • Let $c$ be point in $[a, b]$ where $f$ attains max or min value
• If $c$ is endpoint ($a$ or $b$), then by condition $f(a) = f(b)$, function attains both max and min at endpoints
  • In this case, $f$ is constant on $[a, b]$, thus $f'(x) = 0$ for all $x$ in $(a, b)$ (flat horizontal line)
• If $c$ is interior point of $(a, b)$, then by Fermat's Theorem, $f'(c) = 0$
  • Fermat's Theorem states if function $f$ has local extremum at point $c$ and $f$ is differentiable at $c$, then $f'(c) = 0$ (slope is zero at top of hill or bottom of valley)
• In either case, there exists point $c$ in $(a, b)$ such that $f'(c) = 0$, proving Rolle's Theorem (guaranteed a point with zero slope)

Applications of Rolle's theorem

• Given function $f$ satisfying Rolle's Theorem conditions on interval $[a, b]$, there exists at least one point $c$ in $(a, b)$ such that $f'(c) = 0$ (point with horizontal tangent line)
• To find points where derivative equals zero:
  1. Verify function satisfies Rolle's Theorem conditions on given interval
  2. Find derivative of function, $f'(x)$
  3. Set derivative equal to zero, $f'(x) = 0$, and solve for $x$
  4. Check if solutions lie within open interval $(a, b)$
• Points $x$ within open interval $(a, b)$ that satisfy $f'(x) = 0$ are points where derivative equals zero (locations of horizontal tangent lines, max/min points)

Evaluating functions for Rolle's theorem

• To determine if function $f$ satisfies Rolle's Theorem conditions on interval $[a, b]$, check:
  1. Continuity: Verify $f$ is continuous on $[a, b]$
     • Use definition of continuity or properties of continuous functions (no gaps, holes, jumps, asymptotes)
  2. Differentiability: Verify $f$ is differentiable on $(a, b)$
     • Check if $f$ is polynomial, rational, exponential, logarithmic, or trigonometric function (differentiable on their domains)
     • If $f$ is piecewise, check differentiability at endpoints of each piece and points where pieces connect
  3. Equal function values at endpoints: Verify $f(a) = f(b)$
     • Evaluate function at endpoints $a$ and $b$ and check if values are equal (graph forms a loop)
• If all three conditions are met, function $f$ satisfies Rolle's Theorem conditions on interval $[a, b]$ (guaranteed a point $c$ where $f'(c) = 0$)