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!!
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!
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, or 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 and at endpoints of the domain. From the calculated values of f(x), determine the local/global extrema.
To find the intervals on which a function is increasing and decreasing, you can follow these steps:
1. Start by finding the first derivative of the function, g'(x), which represents the rate of change of the function at each point. In this case, you have g(x) = x√((8-x)^2). To find g'(x), you'll need to apply the chain rule and power rule.
2. Simplify g'(x) by applying the power rule and the chain rule. The derivative of x√((8-x)^2) can be found using the product and chain rule. The result should be g'(x) = √((8-x)^2) - x(8-x)/√((8-x)^2).
3. Now, set g'(x) equal to 0 and solve for x to find the critical points where the slope changes sign.
4. To determine the intervals where the function g(x) is increasing and decreasing, you'll need to analyze the sign of g'(x) in the intervals between the critical points. For example, if g'(x) > 0, the function is increasing, and if g'(x) < 0, the function is decreasing.
Regarding your other question, let's clarify the difference between local max/min and absolute max/min:
- A local maximum (max) occurs at a point where a function reaches the highest value within a small interval (open or closed interval) around that point. This means that there are no higher values for the function in its immediate vicinity, but there may be higher values elsewhere.
- An absolute maximum (max) is the highest point on the entire domain of the function. It is the largest value that the function can achieve, whether it occurs at an endpoint of the domain or a critical point.
- Similarly, local and absolute minima (min) represent the lowest points within a small interval and the entire domain, respectively.
Regarding your last question, it's possible to have multiple local maxima or minima. These occur when the function has several peaks or troughs within its domain. However, there can only be one absolute maximum and one absolute minimum on the entire domain of the function.
I hope this helps! Let me know if you have any further questions.