Calculus Workbook: Maximum & Minimum Values

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Section 3.1 Maximum & Minimum Values

Objectives

Define and identify absolute and local extrema graphically
State and describe the use of the Extreme Value Theorem and Fermat’s Theorem; give examples of when these theorems fail and explain why they fail
Identify critical numbers of a function algebraically and graphically
Use the Closed Interval Method to identify absolute extrema on a closed, bounded interval

Subsection 3.1.1 Before Class

https://mymedia.ou.edu/media/3.1-1.mp4/1_ww51e1ib

Figure 21. Pre-Class Video 1

https://mymedia.ou.edu/media/3.1-2/1_43uuyr3l

Figure 22. Pre-Class Video 2

Subsubsection 3.1.1.1 Definitions

Definition 3.1.1. Absolute (Global) Extrema.

Let \(c\) be a number in the domain \(D\) of a function \(f\text{.}\) Then, \(f(c)\) is the absolute maximum value of \(f\) on \(D\) if \(f(c)\geq f(x)\) for all \(x\in D\text{;}\) \(f(c)\) is the absolute minimum value of \(f\) on \(D\) if \(f(c)\leq f(x)\) for all \(x\in D\text{.}\)

Absolute maxima and minima are also called global maxima/minima (or extrema).

Example 3.1.2.

Below are two graphs. Identify their absolute maxima/minima, if any exist. Justify your responses.

$The graph of \(f(x) = x^2-2\) on the interval \([-2,2]\)$

$The graph of \(f(x) =\cos x\) on the interval \([-7.5,7.5]\)$

In the first picture, the vertex is an absolute minimum. In the second, each peak (\(-2\pi, 0, 2\pi\)) is an absolute maximum, and each valley (\(-\pi, \pi\)) is an absolute minimum.

Definition 3.1.3. Local Max/Min.

The number \(f(c)\) is a local maximum value of \(f\) if \(f(c)\geq f(x)\) when \(x\) is near \(c\text{.}\) \(f(c)\) is a local minimum value of \(f\) if \(f(c)\leq f(x)\) when \(x\) is near \(c\text{.}\)

Example 3.1.4.

Estimate the inputs of any local or absolute extrema.

$The graph of \(f(x)=6x^4-6x^3-5x^2+5x-1\) on the interval \([-1.4,1.9]\text{.}\)$

$The graph of \(f(x)=6x^4-6x^3-5x^2+5x-1\) on the interval \([-1.4,1.9]\text{.}\) Absolute and local extrema are marked and labeled on the graph.$

Question 3.1.5.

Are local extrema automatically absolute extrema? Are absolute extrema automatically local extrema?

No, local extrema are not automatically absolute extrema. However, an absolute extremum can be a local extremum (as long as it doesn’t occur at the endpoints)

Example 3.1.6.

Identify any local maxes, local mins, absolute maxes, or absolute mins on the graph below.

There are local maxima at \(b,r\text{,}\) a local min at \(d\text{,}\) an absolute max at \(r\text{,}\) and an absolute min at \(a\)

Subsubsection 3.1.1.2 Important Results

In order to use maxima/minima, some results will be helpful.

Theorem 3.1.7. Extreme Value Theorem.

If \(f\) is continuous on a closed interval \([a,b]\text{,}\) then \(f\) attains an absolute maximum value \(f(c)\) and an absolute minimum value \(f(d)\) at some numbers \(c,d\) in \([a,b]\text{.}\)

Question 3.1.8.

Why does the extreme value theorem fail on an open interval?

Take the function \(y=x\) on the interval \((0,1)\text{.}\) There is neither an absolute max nor an absolute min, since the function does not achieve the output at the expected max/min.

Theorem 3.1.9. Fermat’s Theorem.

If \(f\) has a local maximum or minimum at \(c\text{,}\) and if \(f'(c)\) exists, then \(f'(c) = 0\text{.}\)

Question 3.1.10.

Is it true that if \(f'(c) = 0\text{,}\) then \(f\) has a local max or min at \(c\text{?}\)

No. If \(f(x) = x^3\text{,}\) then \(f'(0) = 0\) but the function does not have a local extremum at 0.

Subsection 3.1.2 Pre-Class Activities

Example 3.1.11.

Give the coordinates for the absolute and local extrema for the function given below.

The absolute max is at \((4,5)\text{.}\) There is no absolute min. There are local maxes at \((4,5)\) and \((6,4)\text{,}\) and local minima at \((2,2)\) and \((5,3)\)

Example 3.1.12.

Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum.

Answers vary

Example 3.1.13.

Can we apply Fermat’s Theorem to the function \(f(x) = |x|\) on the interval \([-5,5]\text{?}\) Why or why not?

No. \(|x|\) is not differentiable at \(x=0\text{.}\)

Example 3.1.14.

Can we apply the Extreme Value Theorem to the function \(f(x) = |x|\) on the interval \([-5,5]\text{?}\) Why or why not?

Yes. \(f(x)\) is continuous, so we may use the Theorem.

Subsection 3.1.3 In Class

Subsubsection 3.1.3.1 Critical Numbers

Definition 3.1.15. Critical Number/Value/Point.

A critical number of a function \(f\) is a number \(c\) in the domain of \(f\) such that either \(f'(c) = 0\) or \(f'(c)\) does not exist. A critical value is the output at a critical input, namely \(f(c)\text{.}\) A critical point is the coordinate pair \((c, f(c))\text{.}\)

Example 3.1.16.

Find the critical number(s) of \(f(x) = x^3 -3x^2 + 1\text{.}\)

The critical numbers are \(x=0,x=2\text{.}\)

Example 3.1.17.

Find the critical numbers of \(f(t) = t^4 + t^3+t^2 + 1\text{.}\)

The critical number is \(t=0\text{.}\)

Example 3.1.18.

For some function \(f(x)\text{,}\) its derivative is given by \(f'(x) = \dfrac{100(x-2)^2}{5-x^2}\text{.}\) How many critical numbers does \(f\) have? What are they?

\(x=2\) is certainly a critical number. If \(x=\sqrt{5}\) or \(x=-\sqrt{5}\) are in the domain of \(f(x)\text{,}\) then they are critical numbers as well.

Example 3.1.19.

Find the critical numbers of \(f(x) = \cos x\text{.}\)

\(x=\pi k\) for \(k\in \Z\)

Subsubsection 3.1.3.2 Finding Absolute Maxima & Minima

In order to find absolute extrema on closed intervals, we need to find local extrema and compare the values against the endpoints. So, finding absolute maxima and minima comes down to the following process:

Find the critical numbers of \(f\) on a closed interval \([a,b]\text{.}\)
Compute the output values at each critical number.
Compute the output values at the two endpoints.
Compare the results from #2 and #3. The biggest output is the absolute maximum, and the smallest output is the absolute minimum.

Example 3.1.20.

Locate the absolute extrema for \(f(x) = x^3 -3x^2+1\) on the interval \(\left[\dfrac{1}{2},4\right]\text{.}\)

The absolute max is \((4,17)\) and the absolute min is \((2,-3)\text{.}\)

Example 3.1.21.

Locate the absolute extrema for \(f(x) = \sin x\) on the interval \([-2\pi,2\pi]\text{.}\)

The absolute maxes are at \(\lrpar{\dfrac{\pi}{2},1}\) and \(\lrpar{-\dfrac{3\pi}{2},1}\text{,}\) while the absolute minima are at \(\lrpar{-\dfrac{\pi}{2},-1}\) and \(\lrpar{\dfrac{3\pi}{2},-1}\)

Example 3.1.22.

Locate the absolute extrema for the function \(f(t) = (t^2-4)^3\) on \([-3,3]\text{.}\)

The absolute maxes are at \((3,125)\) and \((-3,125)\text{,}\) and the absolute min is at \((0,-64)\text{.}\)

Example 3.1.23.

Find and classify the extrema of the function \(f(x) = x + \dfrac{1}{x}\) on \([0.2,4]\text{.}\)

The absolute max is at \((0,2,5.2)\) and the absolute min is at \((1,2)\text{.}\)

Subsection 3.1.4 After Class Activities

Example 3.1.24.

If \(f(x) = 3x^4 - 4x^3 - 12x^2 + 1\text{,}\) find the absolute maximum and absolute minimium on the interval \([-2,3]\text{.}\)

The absolute max is at \((-2,33)\) and the absolute min is at \((2,-31)\)

Example 3.1.25.

Find the critical numbers of the function \(h(p) =\dfrac{p-1}{p^2+4}\)

\(p = 1\pm \sqrt{5}\)

Example 3.1.26.

If \(a,b\) are positive integers, find the maximum value of \(f(x) = x^a(1-x)^b\text{,}\) for \(0\leq x\leq 1\text{.}\)

The max value is \(f\lrpar{\dfrac{a}{a+b}} = \lrpar{\dfrac{a}{a+b}}^a\lrpar{1-\dfrac{a}{a+b}}^b\)

Example 3.1.27.

Find the absolute maximum and absolute minimum of the function \(f(x) = x\sqrt{3-x}\) on \([-6,3]\text{.}\)

The absolute max is at \(\lrpar{\dfrac{3}{2},\dfrac{3}{2}\sqrt{\dfrac{3}{2}}}\)

Example 3.1.28.

Sketch the graph of a function with two local maxima, one local minimum, and no absolute minimum.

Answers vary

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