Learning Outcomes

  1. Define absolute extrema and local extrema
  2. Find the critical points of a function over a closed interval.
  3. Use critical points to locate absolute extrema over a closed interval.


  • Chapter 4.3  Maxima and Minima

Textbook Assignment

  • p. 376:    108, 110, 113, 119, 122, 124

WeBWork Assignment

  • Application-Extrema

Exit problems of the session 

  1. Find the critical point in the domain of the following functions:

    (a).  $y=4\sqrt{x}-x^2$    (b).   $y=\cos(2x)$   
  2. Find the absolute maxima and absolute minima of the following function over the given interval. 
    (a).  $y=x-x^2$ over  $[-1. 1]$    (b).   $y=x+\sin x$  over  $[0,  2\pi]$


Key Concepts

  • A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum. 
  • Let $c$ be an interior point in the domain of $f(x)$. We say that $c$ is a critical point of $f$ if $f'(c)=0$ or $f'(c)$ is undefined.
  • If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point. 
  • A continuous function over a closed interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint. 
  • Steps of locating absolute extrema of $f(x)$ over a closed interval $[a, b]$:
    1. Evaluate $f$ at the end point $x=a$ and $x=b$.
    2. Find all critical points of $f$ that lie over $(a, b)$. 
    3. Evaluate $f$ at those critical points.
    4. Compare all the values found in 1 and 3. The largest of these values is the absolute maximum of $f$, and the smallest of these values is the absolute minimum of $f$. 


Videos and Practice Problems of Selected Topics

  1. $\rhd$ Introduction to minimum and maximum points (5:29) Find the absolute and relative extrema on a graph.
  2. $\rhd$ Critical points (7:52) The “critical points” of a function are defined followed by a discussion on their relationship with the extrema of the function.
  3. $\rhd$ Finding critical points (5:50) Find the critical points of $f(x) = xe^{-2x^2}$.
  4. * Practice Find the critical points. (4 problems)
  5. $\rhd$ Extreme Value Theorem (7:57) A discussion on the Extreme Value Theorem.
  6. $\rhd$ Finding absolute extrema on a closed interval  (6:55) Find the maximum value of $f(x) = 8\ln x-x^2$ over $[1,4]$.
  7. * Practice Absolute minima and maxima over closed intervals. (4 problems)

STEM Applications