Question

I don't understand how to do this?
I don't know the steps on how to do it! Can you please just tell me the steps?

Find the intervals on which the function is increasing and decreasing.
g(x) = x√((8-x)^2)

(also, what is the difference between local max/min and absolute max/min and can you have two local maximums(/mins) ?

thanks!!

Answer 1

CORRECTION!
I read the problem wrong
..

its actually
g(x) = x√(8 - x^2)

the second x is the only thing that is squared not the whole (8-x).

sorry!

Answer 2

The maxima/minima are part of the important ingredients when we try to sketch the graph of a function without having to plot every possible point.

A local maximum of a function occurs at an interior point c (i.e. f(c+) and f(c-) exist) of its domain if f(x) ≤ f(c) for all x in some open interval containing c. The definition of local minimum is similar.

It is possible to have multiple local maxima/minima.

A global (absolute) maximum of a function occurs at a point c if f(x) ≤ f(c) for all x on its domain. the definition for global minimum is similar.

For example, the function y=x² has a local minimum at x=0. We note that y(0)=0, and at both x=0+ and at x=x-, the value of the function is greater than 0. Therefore x=0 is a local minimum for y=x².

However, there is no local (nor global) maximum for y=x², since for any value of y(x) corresponding to a particular value of x, we can find a greater value of y(x). Since x=±∞ is not in the domain of y=x², there is no local nor global maximum.

To find the local and global maxima/minima, it is necessary to follow these steps:

1. find the domain of the function.
2. Calculate all the critical points in the domain. A critical point is a point where the derivative f'(x) becomes zero, <i>or</i> or where f'(x) is undefined.
3. Since a local extremum can only occur at the following points:
a. interior point where f'(x)=0.
b. endpoints of the domain of f(x).
c. interior points where f'(x) is undefined.
So calculate the values of f(x) at all the critical points <i>and</i> at endpoints of the domain. From the calculated values of f(x), determine the local/global extrema.