1 Maxima and minima

Consider the curve

$y=f\left(x\right)a\le x\le b$

shown in Figure 7:

Figure 7

By inspection we see that there is no $y$ -value greater than that at $x=a$ (i.e. $f\left(a\right)$ ) and there is no value smaller than that at $x=b$ (i.e. $f\left(b\right)$ ). However, the points on the curve at $x_0$ and $x_1$ merit comment. It is clear that in the near neighbourhood of $x_0$ all the $y$ -values are greater than the $y$ -value at $x_0$ and, similarly, in the near neighbourhood of $x_1$ all the $y$ -values are less than the $y$ -value at $x_1$ .

We say $f\left(x\right)$ has a global maximum at $x=a$ and a global minimum at $x=b$ but also has a local minimum at $x=x_0$ and a local maximum at $x=x_1$ .

Our primary purpose in this Section is to see how we might locate the position of the local maxima and the local minima for a smooth function $f\left(x\right)$ .

A stationary point on a curve is one at which the derivative has a zero value. In Figure 8 we have sketched a curve with a maximum and a curve with a minimum.

Figure 8

By drawing tangent lines to these curves in the near neighbourhood of the local maximum and the local minimum it is obvious that at these points the tangent line is parallel to the $x$ -axis so that

$\frac{df}{dx}{\left|\right.}_{x_0}=0$

However, be careful! A stationary point is not necessarily a local maximum or minimum of the function but may be an exceptional point called a point of inflection, illustrated in Figure 9.

Figure 9

Example 2

Sketch the curve $y={\left(x-2\right)}^2+2$ and locate the stationary points on the curve.

Solution

Here $f\left(x\right)={\left(x-2\right)}^2+2$ so $\frac{df}{dx}=2\left(x-2\right)$ .

At a stationary point $\frac{df}{dx}=0$ so we have $2\left(x-2\right)=0$ so $x=2$ . We conclude that this function has just one stationary point located at $x=2$ (where $y=2$ ).

By sketching the curve $y=f\left(x\right)$ it is clear that this stationary point is a local minimum .

Figure 10

Task!

Locate the position of the stationary points of $f\left(x\right)=x^3-1.5x^2-6x+10$ .

First find $\frac{df}{dx}$ :

Answer

Answer

Sketch the curve $y=cos2x0.1\le x\le \frac{3\pi }{4}$ and on it locate the position of the global maximum, global minimum and any local maxima or minima.

Answer