## D8 Maxima and minima

How do you find the maximum (highest) or minimum (lowest) value of a curve? The maximum or minimum values of a function occur where the derivative is zero. That is where the graph of the function has a horizontal tangent.

If you go looking for the horizontal tangents (i.e. where the derivative = 0), you will be able to pinpoint the maxima or minima of a curve.

Using calculus we can find the derivative of a function $$f\left(x\right)$$ with respect to $$x$$ and use this to find maximum and minimum values of $$f\left(x\right)$$ and the values of $$x$$ where they occur. We can therefore use calculus to solve problems that involve maximizing or minimizing functions.

### 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*}

#### Second Derivative Test.

Let $$x$$ satisfy $$\left(1\right)$$ then if \begin{align*} f^{"}\left(x\right) & \begin{cases} =0,\text{ x is an inflection point}\\ <0\text{, f\left(x\right) is a local maximum}\\ >0\text{, f\left(x\right) is a local minimum.} \end{cases} \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 $$\left(2.1\right)$$1 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?

2. $$\left(3,-28\right)$$
3. $$144\textrm{cm}^{2}$$
4. $$750$$