Can relative maximum and minimum exist when the derivative....

• MidgetDwarf
In summary, the concept of relative maximum and minimum points is not dependent on the existence of a derivative. A function can have an absolute maximum or minimum at a point where the derivative does not exist. However, the derivative can help in finding extrema points. The function must exist at a point in order for it to have a maximum or minimum at that point.
MidgetDwarf
Can relative maximum and minimum points exist when a function is defined at say x=c, however the derivative does not exist or tends to infinity? Ie the graph of. F (x)= |x|, for x=c=o. If I am correct the relative minimum is at o, can it also be the abs minimum?

I recalled the theorem by memory.

Let the function f be defined on the closed interval from a to b and have a relative maximum or minimum at x=c, where c is in the open interval from a to b.

If the derivative exist as a finite number at x=c, then the derivative is o at that point.

I went through the derivation of the proof, and the theorem (by analyzing my derivation), does not say what happens when.

1) when the derivative fails to exist at a point c.
2) at the endpoints, or that there is always a max or min when the derivative equals 0.

I sometimes tutor my little neighbor next door for free, because it helps me practice things I havnt used in a while. Today I told him, that a relative max or min can occur if the function is defined at that point, does not have to be necessary differential at that point. I use y=|x| as an example.

I felt bad, because I believed I gave him the wrong answer. Sorry for the long post.

The definitions of absolute and relative maxima and minima have nothing to do with derivatives:

$f: [a,b] \to \mathbb{R}$ has an absolute maximum (minimum) at $c \in [a,b]$ if and only if $f(c) \geq f(x)$ ($f(c) \leq f(x)$) for all $x \in [a,b]$.

$f: [a,b] \to \mathbb{R}$ has a relative maximum (minimum) at $c \in [a,b]$ if and only if there exists a $\delta > 0$ such that for all $x \in [a,b]$, if $|x - c| < \delta$ then $f(c) \geq f(x)$ ($f(c) \leq f(x)$).

Thus $f(x) = |x|$ has a absolute minimum at x = 0; this is obvious from the fact that $|x| \geq 0$ for all $x \in \mathbb{R}$.

There is, of course, a theorem which says that if a function is differentiable at a relative extremum then its derivative must vanish there. But the converse does not hold: $f: [-1,1] \to \mathbb{R} : x \mapsto x^3$ has $f'(0) = 0$ but zero is neither a maximum nor a minimum of any type.

This theorem can help you find extrema, but you must always check that a point where the derivative vanishes really is a relative extremum, and you must also check whether the end points and any points where the derivative does not exist are extrema.

MidgetDwarf
pasmith said:
The definitions of absolute and relative maxima and minima have nothing to do with derivatives:

$f: [a,b] \to \mathbb{R}$ has an absolute maximum (minimum) at $c \in [a,b]$ if and only if $f(c) \geq f(x)$ ($f(c) \leq f(x)$) for all $x \in [a,b]$.

$f: [a,b] \to \mathbb{R}$ has a relative maximum (minimum) at $c \in [a,b]$ if and only if there exists a $\delta > 0$ such that for all $x \in [a,b]$, if $|x - c| < \delta$ then $f(c) \geq f(x)$ ($f(c) \leq f(x)$).

Thus $f(x) = |x|$ has a absolute minimum at x = 0; this is obvious from the fact that $|x| \geq 0$ for all $x \in \mathbb{R}$.

There is, of course, a theorem which says that if a function is differentiable at a relative extremum then its derivative must vanish there. But the converse does not hold: $f: [-1,1] \to \mathbb{R} : x \mapsto x^3$ has $f'(0) = 0$ but zero is neither a maximum nor a minimum of any type.

This theorem can help you find extrema, but you must always check that a point where the derivative vanishes really is a relative extremum, and you must also check whether the end points and any points where the derivative does not exist are extrema.

Thanks. This is exactly what I was thinking. The requirement is that the function needs to exist at that point? That is why I used y=|x|, thanks for the insight and fast response, I now feel better I did not give him miss information.

The function has to exist, sure.
There are useful statements about the maxima and minima of functions that are differentiable at one point or in some interval, but the concept of maxima and minima is much more general and can be used for non-differentiable functions as well.

For a contrived example of MFB's post, take ##f=\chi \mathbb Q## (Take it, please!) nowhere-differentiable (since, e.g., nowhere continuous) , with Max reached at ##\mathbb Q## and min reached at ##\mathbb R-\mathbb Q##.

Question 1: Can relative maximum and minimum exist when the derivative is undefined?

Yes, it is possible for a function to have a relative maximum or minimum when the derivative is undefined. This typically occurs at points where the function has a sharp corner or a vertical tangent line.

Question 2: Is it possible for a function to have a relative maximum or minimum at a point where the derivative is zero?

Yes, a function can have a relative maximum or minimum at a point where the derivative is zero. This is known as a critical point, and it can occur at local extrema or points where the function changes from increasing to decreasing (or vice versa).

Question 3: How can I determine if a point is a relative maximum or minimum using the derivative?

If the derivative is positive at a point, the function is increasing and the point is a relative minimum. If the derivative is negative at a point, the function is decreasing and the point is a relative maximum. If the derivative is zero at a point, further analysis is needed to determine if it is a relative maximum or minimum.

Question 4: Can a function have multiple relative maxima or minima?

Yes, a function can have multiple relative maxima and minima. This occurs when the derivative changes sign at multiple points, indicating a change in the direction of the function.

Question 5: How does the graph of a function relate to the existence of relative maxima and minima?

The graph of a function can visually show the existence of relative maxima and minima by the presence of peaks or valleys. These correspond to points where the derivative is zero or changes sign, indicating a change in the direction of the function.

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