Increasing, Decreasing, Critical Points, and Relative Extrema

Back to main page

Today’s plan

[greybox2]

  • Warm-up and Overview of Increasing/Decreasing
  • iClicker
  • Lecture Problem 1a, Section 4.1.
  • Warm-up and Overview of Critical Points and Relative Extrema
  • Lecture Problem 3, Section 4.1. [/greybox2]

Midterm and Gateway return

If you were not in class on Wednesday when I passed out the exams, please check the instruction I put on the implicit differentiation page. I will pass them out again today along with the Gateway exams.

Gateway retakes

If you need retake the Gateway, please check the Canvas announcement. If you would like to discuss with me what you need to work on or need help with anything, please send me an email! :)

If you were not here when we did related rates, spend some time after class today looking at the related rates page.

Warm-up and Overview of Increasing/Decreasing

There is a lot of stuff to cover today so I want us to get a quick warm-up on stuff we’re doing today. One thing you see Dr. Browne talk about is the idea of increasing/decreasing.

[greybox2] Definition. A function $f$ is:

  • Increasing on an interval $(a, b)$ if whenever $x_1 < x_2$ then $f(x_1) < f(x_2)$ on $(a, b)$.
  • Decreasing on an interval $(a, b)$ if whenever $x_1 < x_2$ then $f(x_1) > f(x_2)$ on $(a, b)$.

WARNING: In many other resources, this is more specifically refered to as “strictly increasing/decreasing”. [/greybox2]

[greybox] Warm-up 1. The definition that Dr. Browne gives in her video is a very formal definition of increasing and decreasing. To make sure you understand what that statement intuitively means, fill in the following blank and explain your reasoning.

A function $f$ is increasing on the interval $(a, b)$ if $f(x)$ gets ________ as $x$ goes from left to right in the interval $(a, b)$.

Now repeat this problem but replace “increasing” with “decreasing”. [/greybox]

As it turns out, if our function is differentiable, then its derivative can be very helpful in telling us exactly when $f$ is increasing. This is the theorem that Dr. Browne gave you in her video:

[greybox2] Theorem.

  1. If $f’(x) > 0$ on $(a, b)$, then $f$ is increasing on $(a, b)$.
  2. If $f’(x) < 0$ on $(a, b)$, then $f$ is decreasing on $(a, b)$.
  3. If $f’(x) = 0$ on $(a, b)$, then $f$ is constant on $(a, b)$. [/greybox2]

[greybox] Warm-up 2. Recall that $f’(a)$ is the slope of the line tangent to the graph of $f$ at $a$. Experiment with this example in Desmos (especially play around with the slider!):

https://www.desmos.com/calculator/hkqawhaajm

  1. What do you notice about $f’(x)$ (i.e. the slope of the tangent line) when $f$ is increasing? Decreasing?
  2. Explain why your results are consistent with the theorem. [/greybox]

iClicker

Your favorite. :)

Lecture Problem 1a, Section 4.1

Recall Dr. Browne’s 3-step process for finding intervals of increasing/decreasing is the following:

  1. Find $x$-values when $f’(x) = 0$ or $f’(x)$ is discontinuous.
  2. Extract open intervals around these $x$-values.
  3. Sign Test: Pick a test point in each interval and find the sign of the first derivative $f’(x)$.
    • If $f’(c) > 0$, then $f$ is increasing on the interval.
    • If $f’(c) < 0$, then $f$ is decreasing on the interval.

Use this 3-step process to approach the following problem and clearly outline which step is which in your work.

[greybox] Lecture Problem 1a, Section 4.1. Let $f(x) = x^3 - 3x^2 + 5$.

(a) Find the intervals where $f(x)$ is increasing and decreasing. [/greybox]

[solution] $f$ is increasing on $(-\infty, 0) \cup (2, \infty)$ and decreasing on $(0, 2)$. [/solution]

Warm-up and Overview of Critical Points and Relative Extrema

Recall that relative maximum/minimum has the rather technical definition:

[greybox2] Definition. A function $f$ has a:

  • Relative maximum at $x = c$ if there exists an open interval $(a, b)$ containing $c$ such that $f(c) > f(x)$ for all $x$ in $(a, b)$.
  • Relative minimum at $x = c$ if there exists an open interval $(a, b)$ containing $c$ such that $f(c) < f(c)$ for all $x$ in $(a, b)$. [/greybox2]

[greybox] Warm-up 3. Come up with a more intuitive way of describing a relative maximum/minimum. Also, why does a relative maximum/minimum NOT need to be a maximum/minimum for the entire function? [/greybox]

As for how we actually find relative maximum/minimum, we rely on the idea of critical points. In case you forgot the definition, here it is:

[greybox] Definition. A critical point of a function $f$ is a number $x$ in the domain of $f$ such that

\[\begin{align*} f'(x) = 0 \qquad\text{ OR }\qquad f'(x) \text{ is discontinuous}. \end{align*}\]

[/greybox]

[greybox2] Warm-up 4: Critical point do NOT always give a relative maximum/minimum. An example of where critical point does not lead to a relative maximum/minimum is the critical point $x = 0$ of $f(x) = x^3$. Verify that $x = 0$ is, in fact, a critical point yet $f(0)$ is NOT a relative maximum or minimum. [/greybox2]

Lecture Problem 3, Section 4.1

[greybox2] First Derivative Test.

  1. Find critical points of the function $f(x)$ (call them $c$).
  2. Perform a sign test on $f’$ around the critical points.

Conclusion:

  • If $f’$ changes from positive to negative at $x = c$, then $x = c$ is a relative maximum.
  • If $f’$ changes from negative to positive at $x = c$, then $x = c$ is a relative minimum. [/greybox2]

[greybox] Lecture Problem 3, Section 4.1. Use the first derivative test to find the relative maximum and relative minimum of the function $f(x) = 3x^3 - 18x^2 + 6$. Clearly outline which step is which in your work.

Once you find the relative maximum and minimum, try confirming that your answer is correct by plotting the function in a calculator such as Desmos. Do this AFTER you find the relative maximum/minimum, not before. [/greybox]

[solution] Relative maximum is $6$ and relative minimum is $-90$. [/solution]