More on Increasing, Decreasing, Critical Points, and Relative Extrema

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Today’s plan

[greybox2]

  • Overview of Increasing/Decreasing
  • iClicker
  • Problem on Increasing/Decreasing
  • Overview of Critical Points and Relative Extrema
  • Lecture Problem 3, Section 4.1. [/greybox2]

Gateway retakes

If you need retake the Gateway, please check the Canvas announcement. Today is the last day of this week that Gateway retakes are being offered (there will be more later).

If you would like to discuss with me what you need to work on or need help with anything, please send me an email with an availability you would like to meet! :)

Overview on Increasing/Decreasing

Last class session, I asked you to come up with an intuitive way to understand the idea of “increasing/decreasing” via the following problem. Please do this again – we’ll discuss how to think of this graphically as well in class.

[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 $f(x)$ always gets ______________ as $x$ goes from left to right in the interval $(a, b)$.”

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

[greybox] IMPORTANT: Notice that

  1. Increasing/decreasing on its own has NOTHING to do with the derivative of $f$.
  2. Increasing/decreasing has NOTHING to do with whether or not $f$ itself is positive/negative/zero. [/greybox]

As we mentioned last time, the derivative of your function can be used to investigate whether or not a function is increasing/decreasing in a small neighborhood of your point. We will discuss this one as well.

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

https://www.desmos.com/calculator/mqyx3ipiyi

What do you notice about $f$ (i.e. the slope of the tangent line to graph of $f$ at $x = c$) when $f’(c)$ is positive? Negative? Zero? [/greybox]

If you did the problem above, then you should understood why the theorem Dr. Browne gave you below is true:

[greybox2] Theorem. Let $f$ be differentiable on $(a, b)$.

  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]

iClicker

Your favorite! :D

Problem on Increasing/Decreasing

[greybox] Problem 1. Let

\[\begin{align*} f(x) = x^3 - 6x^2 + 1. \end{align*}\]

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

[solution]

  • Increasing: $(-\infty, 0) \cup (4, \infty)$.
  • Decreasing: $(0, 4)$. [/solution]

Overview of Relative Extrema and Critical Points

You probably recall Dr. Browne talking a little bit about relative extrema from her video. The definition she gave is pretty formal so let’s make sure you actually understand what is meant.

[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]

If you look at Problem 1 above, you will notice that when $f’(x) = 0$, we have a “peak” or “valley” in the graph. We can use this idea to help us actually figure out relative extrema.

[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]

[greybox] Problem 2. Let $f(x) = \frac{x^2}{x^2 - 4}$. Find the critical points of $f(x)$.

Hint: You may need to look up the quotient rule. [/greybox]

[solution] $x = 0$. [/solution]

As a warning on critical points, you should know that a critical point does NOT always give a relative extremum.

[greybox] 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. [/greybox]

First Derivative Test

[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] Problem 3. Use the first derivative test to find the relative maximum and relative minimum from Problem 1. [/greybox]

[solution] Relative maximum is $1$ and relative minimum is $-31$. [/solution]