Section 4.7 Applied Optimization Problems
Many scientific and engineering questions can be phrased in terms of finding the global minimum or maximum of a function, such as minimizing cost or weight or maximizing the use of limited resources. The solution to such questions often breaks into several steps:
rephrasing a verbal question in terms of a quantity to be minimized, given in terms of other quantities whose values one can adjust in seeking the optimal outcome;
expressing the quantity to optimize as a function of a single variable;
finding the local extrema of this function; and finally
identifying the optimum (global minimum) from amongst these local extrema.
The First and Second Derivative Tests for Global Extrema.
There is bad news and good news in using the ideas of previous section to find a global extremum.
The bad news is that the domain is often not a closed interval for which the Extreme Value Theorem applies. Instead it is often an open interval, such as “any positive value”.
The good news is that there is often a unique critical point, and then the only question is whether it is a global maximum, a global minimum, or neither. Derivatives again answer this question:
Theorem 4.7.1. (The first derivative test for global extrema).
Suppose that function \(f\) has a unique critical number \(c\) on an interval.
If \(f'(x)>0\) to the left of \(c\) (\(x < c\)) and \(f'(x) < 0\) to the right of \(c\) (\(x > c\)), then \(f(c)\) is the maximum value of \(f\) on that interval.
If \(f'(x) < 0\) to the left of \(c\) and \(f'(x)>0\) to the right of \(c\text{,}\) then \(f(c)\) is the minimum value of \(f\) on that interval.
If \(f'(x)\) has the same sign on either side of \(c\) then \(f\) has no global extremum on that interval, except possibly at an endpoint.
Checkpoint 4.7.2.
Make sketches illustrating each case.
Theorem 4.7.3. (The second derivative test for global extrema).
Suppose that function \(f\) has a unique critical number \(c\) on an interval.
Note: if \(f''(c)=0\) or DNE, other tests must be used, like the one above.
Checkpoint 4.7.4.
Make sketches illustrating each case.
A Strategy for Optimization Problems.
The approach blends some elements from related rates problems with ideas from this chapter.
Read the question carefully. (Familiar?) Note all the relevant information.
If appropriate, draw a diagram to summarize this information.
Name all relevant quantities, in particular one to be optimized (let us call it \(Q\)) and others whose values can be adjusted (say \(x\text{,}\) \(y\text{,}\) etc.)
Find a formula for the quantity to be optimized in terms of the other quantities: say \(Q(x,y,\dots\)).
If this formula involves more than one variable, seek equations relating these, and use them to eliminate all but one independent variable, giving the quantity to be optimized as a function of a single variable; say \(Q(x)\text{.}\)
Determine the allowable values of the independent variable[s], and thus determine the domain of the above function (\(Q(x)\)).
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Using the various derivative tests above, or otherwise, find the global optimum value and the values of the independent variable[s] that give it.
Do not forget that the optimum might occur at an endpoint!
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Answer the Question.
The full answer often involves giving the values of all variables introduced in Step 3, and putting back in physical units.
Exercises Exercises
Study
Calculus Volume 1, Section 4.7 2 ; in particular Examples 33–35 and 37; Checkpoints 31–34 and 36; and a selection from Exercises 311–314, 315–318, 319–321, 322–326, 335–336 and 351–355.
Here the exercises are grouped in ranges by "question type", so start by trying one or two from each of the seven ranges; some suggested selections are Exercises 311, 316, 320, 322, 335 and 353.
openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems
openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems
