Consider the function \(f(x)=x^4-2x^2+1\). Without using too much theory we can express \(f\) in the following way: \[f(x)=\left(x^2-1\right)^2=(x-1)^2\cdot (x+1)^2.\] From this expression we see that the function is always non-negative and its minimum \(0\) is attained at \(2\) points: \(-1\) and \(1\). On the real line, the function does not have maximum, because as \(x\) approaches infinity, the value of \(f\) goes to infinity.

Let us now take a look at the graph of the function \(f\):

The graph shows that the minimum is \(0\) at two points: \(-1\) and \(1\). However, the graph shows that the point \(x=0\) is special: the function has a ``peak’’ at that point. It is not real maximum (because we saw that the function goes to \(+\infty\)). This point is called ``local maximum.’’ We are now going to define this term precisely.

In an analogous way we define the local minimum.

Notice that the previous theorem makes sense: If \(f\) attains maximum or minimum at \(a\), the tangent has to be horizontal at \(a\), hence the slope is \(0\). The previous proof is just a formal way of writing this observation.