First year: Tangent, maximum and minimum

(I)  Tangents

The tangent to the graph of a function at the point is a line such that:

-  its slope is equal to

-  it passes through the point

The equation of the tangent to the graph of a function at the point is given by the following formula:

Example: Find the equation of the tangent to the graph of at the point

We have and, since , we obtain

(II)  Maximum and minimum

A function is said to have a local maximum at if there exists such that, for , we have .

Intuitively, it means that around the graph of will be below .

Similarly, a function is said to have a local minimum at if there exists such that, for , we have .

This time, the graph of will be situated above for values of around

Examples:

.

From the graph, it is rather obvious that the function has a unique minimum and that this minimum is global (i.e. the whole graph is above this minimum).

On the other hand, if we take , the situation is rather different:

Here, we have a local maximum and a local minimum.

Minima and maxima have one thing in common: say has a local minimum at . Then the tangent to the graph of at the point is a horizontal line:

The slope of the tangent is therefore .

Remember, the slope of the tangent to the graph of at the point is equal to so here we end up with .

If has a local minimum or a local maximum at , we therefore have .

In general, the solutions of are called stationary points. There are three different kinds of stationary points: local minima, local maxima and turning points.

You can classify them as follows:

Say is a stationary point. Then if

-  , there is a local maximum at .

-  , there is a local minimum at .

-  , there is a turning point at .

Example: . Find and classify the stationary points of .

To find the stationary points, we solve :

Here, , so that .

Next, we calculate and use the rule above to classify the stationary points:

.

, so that has a local minimum at

, so that has a local maximum at .

Let’s have a look at the graph of :

The graph indicates that there is indeed a local minimum at and a local maximum at . The graph also indicates that they are both local and not global.