Maximum and Minimum of a Function

Local maximum and minimum

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 \( f \):

graph of a function

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.

Definition (Local maximum)
 
The function \( f \) attains a local maximum at point \( a \) if there is an interval of the form \( (p,q) \) that contains \( a \) (i.e. \( p< a< q \)) such that \( f(x)\leq f(a) \) for all \( x\in (p,q) \).

In an analogous way we define the local minimum.

Definition (Local minimum)
 
The function \( f \) attains a local minimum at point \( a \) if there is an interval of the form \( (p,q) \) that contains \( a \) (i.e. \( p< a< q \)) such that \( f(x)\geq f(a) \) for all \( x\in (p,q) \).

Application of differential calculus in finding the local extrema

Theorem
 
If the function \( f \) attains local minimum or local maximum at the point \( a\in\mathbb R \), and if \( f \) is differentiable at \( a \), then \( f^{\prime}(a)=0 \).

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.


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