Learning Lab

Definition

The maximum or minimum values of a function \(f\left(x\right)\) occur when the derivative \[\begin{align*} f'\left(x\right) & =0. & \left(1\right) \end{align*}\]

Example 1

The distance \(s\,km,\) to the nearest \(km\), of a fishing boat from port at any time, \(t\) hours, is given by the formula \[\begin{align*} s & =2+8t-2.5t^{2}. \end{align*}\] When is the boat furthest from port and what is its distance from the port at that time?

Solution

For a maximum or minimum, \(ds/dt=0.\) That is

\[\begin{align*} \frac{ds}{dt} & =8-5t\\ & =0. \end{align*}\] So, \[\begin{align*} 8-5t & =0\\ 5t & =8\\ t & =1.6\text{ hours.} \end{align*}\] Now this could be a maximum or minimum distance. However, \[\begin{align*} \frac{d^{2}s}{dt^{2}} & =-5\\ & <0. \end{align*}\] Hence \(t=1.6\) is a maximum. When \(t=1.6\) hours, the distance from port, \[\begin{align*} s & =2+8(1.6)-2.5\left(1.6\right)^{2}\\ & =8.4\,km. \end{align*}\] The boat is furthest from port after \(1.6\) hours and the distance from port, at that time, is \(8.4\,km\).

Example 2

Find the maximum product of two numbers that have a sum of \(10.\)

Solution

Let the numbers be \(a\) and \(b\). Then \[\begin{align*} a+b & =10. & \left(2.1\right) \end{align*}\] Let the product of the two numbers be \(P\) so \[\begin{align*} P & =a\times b. \end{align*}\] From \(\left(2.1\right)\)11 We need to get \(P\) in terms of \(a\) or \(b\) so that we take a derivative like \(dP/da\) or \(dP/db\). In this case we write \(P\) as a function of \(b\) but identical results are obtained is we make \(P\) a function of \(a.\) \[\begin{align*} a & =10-b \end{align*}\] so \[\begin{align*} P & =\left(10-b\right)b\\ & =10b-b^{2}. \end{align*}\] Now \[\begin{align*} \frac{dP}{db} & =10-2b. \end{align*}\] For a maximum or minimum, \(dP/db=0,\) \[\begin{align*} 10-2b & =0\\ 2b & =10\\ b & =5. & \left(2.2\right) \end{align*}\] But \[\begin{align*} \frac{d^{2}P}{db^{2}} & =-2\\ & <0 \end{align*}\] and we have a maximum. Substituting \(b=5\) in \(\left(2.1\right)\) we find \(a=5.\)

Hence the two numbers adding to \(10\) and having a maximal product are \(a=b=5\) and the maximum product is \(25.\)

Example 3

Find the minimum value of the function \(f\left(x\right)=x^{2}-5x+6\).

Solution

We have \[\begin{align*} f'\left(x\right) & =2x-5. & \left(3.1\right) \end{align*}\] For a maximum or minimum we know

\[\begin{align*} f'\left(x\right) & =0\\ 2x-5 & =0\\ 2x & =5\\ x & =\frac{5}{2}. \end{align*}\] Since \[\begin{align*} f^{"}\left(x\right) & =2\\ & >0 \end{align*}\] for all values of \(x\) we know we have a minimum. Hence the minimum value of \(f\left(x\right)=x^{2}-5x+6\) occurs at \(5/2.\)

The minimum value of the function is \[\begin{align*} f\left(x\right) & =\left(\frac{5}{2}\right)^{2}-5\left(\frac{5}{2}\right)+6\\ & =\frac{25}{4}-\frac{25}{2}+\frac{12}{2}\\ & =\frac{25}{4}-\frac{50}{4}+\frac{24}{4}\\ & =-\frac{1}{4}. \end{align*}\]

Exercises

1. Find two positive numbers whose sum is 18 such that the sum of their squares is a minimum.

2. Find the turning point of the parabola defined by \(y=f(x)=5x^{2}-30x+17\).

3. What is the maximum area that can be enclosed if a rectangle is created with a piece of wire \(48\,cm\) long?

4. The annual profit \(P\) made on a garment is related to the number \(n\) that are produced by the formula \(P(n)=300n-7200-0.2n^{2}.\) How many garments should be produced to maximize profit?