Local extrema versus maximum and minimum

A local extremum and a maximum or minimum are different things¹. We see the difference with $f(x) = x^4 - 2x^2$ .

Finding the extrema

Since $f'(x) = 4x^3 - 4x = 4x(x - 1)(x + 1)$ , there are three points where $f' = 0$ .

$x$	$\cdots$	$-1$	$\cdots$	$0$	$\cdots$	$1$	$\cdots$
$f'(x)$	$-$	$0$	$+$	$0$	$-$	$0$	$+$
$f(x)$	decreasing	local min $-1$	increasing	local max $0$	decreasing	local min $-1$	increasing

The local maximum is $0$ and the local minimum is $-1$ .

A local maximum is not a maximum

The important point here is that the local maximum is not the greatest value of the whole graph. Since $f \to \infty$ as $x \to \pm\infty$ , this function has no maximum at all. The point $(0, 0)$ is merely the highest in its own neighborhood.

A minimum, on the other hand, does exist. Both local minima equal $-1$ , and that is the lowest value anywhere, so the minimum is $-1$ . There is no harm in two different values of $x$ attaining it.

Item	Local extremum	Maximum and minimum
Range compared against	a neighborhood of the point	the whole domain, or an interval
Values for this function	local max $0$ , local min $-1$	no maximum, minimum $-1$

A local extremum is a comparison made nearby; a maximum or minimum is a comparison made overall. That is why finding a maximum or minimum means examining not only the points with $f' = 0$ but also the ends of the interval and the behavior at infinity.

Restricting the interval changes the answer

On $0 \leq x \leq 2$ the answer is different.

Candidate	$x$	$f(x)$
Local minimum	$1$	$-1$
Left end of the interval	$0$	$0$
Right end of the interval	$2$	$8$

The maximum is $8$ and the minimum is $-1$ . The point $x = 2$ is not an extremum, but as an end of the interval it is a candidate for the maximum.

The shape of the graph

Factoring $f(x) = x^2(x^2 - 2)$ shows that the graph meets the $x$ -axis at $x = 0$ and $x = \pm\sqrt{2}$ . Since $x = 0$ is a double root, the curve touches the axis at the origin without crossing. On $0 < |x| < \sqrt{2}$ we have $f < 0$ , so the graph dips below the axis.

The shape is a W. Being even it is symmetric about the $y$ -axis, with a small hill at the origin and two valleys at $x = \pm 1$ .

The quartic on the graph is $y = x^4 - 2x^2$ , the cubic is the derivative, and the large dots are the local minima $(-1, -1)$ and $(1, -1)$ together with the local maximum $(0, 0)$ .

Maximum and minimum, Wikipedia