Solved Examples: Extreme Value Theorem for Absolute Extrema

📚 Quick Study Guide

• 📈 Extreme Value Theorem (EVT): If a function $f$ is continuous on a closed interval $[a, b]$, then $f$ must attain an absolute maximum and an absolute minimum on that interval.
• 🔍 Critical Points: These are points $c$ in the interval $(a, b)$ where $f'(c) = 0$ or $f'(c)$ does not exist.
• 📝 Steps to Find Absolute Extrema:
  1. Find all critical points of $f$ in $(a, b)$.
  2. Evaluate $f$ at all critical points and at the endpoints $a$ and $b$.
  3. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

Practice Quiz

1. Question 1: Which condition MUST be met for the Extreme Value Theorem to apply to a function $f$ on an interval $[a, b]$?
   1. $f$ must be differentiable on $[a, b]$.
   2. $f$ must be continuous on $[a, b]$.
   3. $f$ must be increasing on $[a, b]$.
   4. $f$ must be decreasing on $[a, b]$.
2. Question 2: Find the critical points of the function $f(x) = x^3 - 6x^2 + 5$ on the interval $[-1, 5]$.
   1. $x = 0, 4$
   2. $x = -1, 5$
   3. $x = 2, -2$
   4. $x = 1, -1$
3. Question 3: Given $f(x) = x^2 - 4x + 6$ on $[0, 3]$, find the absolute minimum value.
   1. 2
   2. 3
   3. 6
   4. 0
4. Question 4: For the function $f(x) = \frac{1}{x}$ on the interval $[1, 4]$, does the Extreme Value Theorem guarantee the existence of absolute extrema?
   1. Yes, because the function is continuous.
   2. No, because the interval is not closed.
   3. Yes, because the interval is closed.
   4. No, because the function is not continuous at $x = 0$, which is outside the interval.
5. Question 5: Find the absolute maximum of $f(x) = x^3 - 3x^2$ on the interval $[-1, 3]$.
   1. 0
   2. -2
   3. 4
   4. -4
6. Question 6: What are the endpoint values to consider when finding the absolute extrema of $f(x)$ on the interval $[-2, 5]$?
   1. -2 and 0
   2. 0 and 5
   3. -2 and 5
   4. -5 and 2
7. Question 7: Find the absolute minimum value of $f(x) = x^2 + 2x - 1$ on the interval $[-2, 1]$.
   1. -2
   2. -1
   3. 2
   4. -3