# Find a function with the same maximum and minimum? What?

## Homework Statement

Name two different functions that share a maximum and minimum at -4.

## The Attempt at a Solution

This is a vague question. I guess horizontal lines would be one. I can't think of another.
A function that is always a point?

Thanks.

Mark44
Mentor

## Homework Statement

Name two different functions that share a maximum and minimum at -4.

## The Attempt at a Solution

This is a vague question. I guess horizontal lines would be one. I can't think of another.
A function that is always a point?
Some textbooks distinguish between maximum and maximum value, and similarly for minimum and minimum value. Here maximum or minimum refer to the x-value, while maximum value or minimum value refer to the function value.

For example, the function f(x) = x2 - 2x has a minimum of 1 and a minimum value of -1.

As I interpret your problem, you need to find one function that has a maximum at -4 and another that has a minimum at -4.

It wouldn't hurt to get clarification from your instructor.

How about f(x) = -4 ?

HallsofIvy
Homework Helper
How about f(x) = -4 ?
I believe that was what zeion said in his original post. The problem is to find another function that has the same number as maximum and minimum.

Zeion, what, exactly, are the conditions on the function? Would "f(0)= -4, f(x) not defined for any other value of x" work?

AGNuke
Gold Member
This question needs two clarifications:

1.) Maxima and Minima or actual values? (I take it as actual values, maxima and minima can't exist on a single point)

2.) Whether the question is demanding a single equation have both maximum (values?) and minimum (values?) at x = -4 or two equation each with maximum and minimum.

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two thoughts:

1) are discontinuous functions allowed?

2) how about a function with a local min equal to a local max?

AGNuke
Gold Member
No local Min or Max for discontinuous functions as I believe. Main is whether the min/max value or minima and maxima (which corresponds to x)

eumyang
Homework Helper
No local Min or Max for discontinuous functions as I believe. Main is whether the min/max value or minima and maxima (which corresponds to x)
int(x) has jump discontinuities at each x ε Z, and yet, at every "step" (look at the graph) every point is considered to be both a local minimum and a local maximum.

AGNuke
Gold Member
Sorry to disappoint you, there's no minima or maxima for a straight line as there is no change in slope.

How can we differentiate a non-continuous function to find out its zero slope?

eumyang
Homework Helper
I'm going by the definitions of "local maximum" and "local minimum" in my precalculus book:
A local maximum of a function f is a value f(c) that is greater than or equal to all range values of f on some open intervale containing c. If f(c) is greater than or equal to all range values in f, then f(c) is the maximum (or absolute maximum) of f.
(The book also has a similar definition for a local minimum)
So, for the function f(x) = int(x), for any a that is NOT an integer, f(a) is both a local maximum and a local minimum of f.