Decreasing Function: Understanding Why g is Decreasing at x=2 and x=-2

In summary, the question is regarding the function g and why it is decreasing when x=2 and x=-2. The answer states that the function is decreasing because its derivative is negative. The derivative can be thought of as a slope, where a negative slope indicates a decreasing function and a positive slope indicates an increasing function. The first derivative test also supports this, stating that if the derivative is negative at each point within the interval, then the function decreases throughout the interval including the endpoints. However, it is unclear why this is the case for x=-2 and x=2 when the derivative is zero. The definition of the first derivative test does not provide an explanation for this.
  • #1
UrbanXrisis
1,196
1
the question is http://home.earthlink.net/~urban-xrisis/clip_image002.jpg

The answer is A but I don't understand why the function g would be decreasing when x=2 and x=-2
 
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  • #2
The answer states between -2 and 2 not only when x=2 or x=-2. The function is decreasing because its derivative is negative. Where a derivative is negative the function is decreasing, where a derivative is positive the function is increasing. Think of your derivative as a slope, a negative slope means your function goes down from left to right(decreasing), and a positive slope means your function goes up from left to right(increasing).
 
  • #3
no, it states between -2 and 2 AND when they are equal. why?
 
  • #4
[tex] -2 \underline{<}x\underline{<}2[/tex]
is different from:
[tex] -2 < x < 2[/tex]

why would the slope be decreasing at x=-2 and x=2 when the derivative is zero?
 
  • #5
I misunderstood your question sorry about that.The first derivative test(straight from a calc book) states:

"Suppose that [tex]f[/tex] is continuous at each point of the closed interval [tex][a,b][/tex] and differentiable at each point of its interior [tex](a,b)[/tex]. if [tex]f'>0[/tex] at each point of *[tex](a,b)[/tex], then [tex]f[/tex] increases throughout *[tex][a,b][/tex].if [tex]f'<0[/tex] at each point of [tex](a,b)[/tex], then [tex]f[/tex] decreases throughout [tex][a,b][/tex]."

*notice that they are using () meaning not including endpoints, however, after they use[] which means that the whole interval is increasing including the end points, this is by definition. As to why I don't remember right now, the calc book isn't helping much either, but I'm pretty sure the definition is right.
 

Related to Decreasing Function: Understanding Why g is Decreasing at x=2 and x=-2

1. What is a decreasing function?

A decreasing function is a type of mathematical function where the output values decrease as the input values increase. This means that as the independent variable (usually denoted as x) increases, the dependent variable (usually denoted as y) decreases.

2. Why is g decreasing at x=2 and x=-2?

The function g is decreasing at x=2 and x=-2 because the slope of the function at these points is negative. This means that as x increases or decreases, the value of g decreases as well. In simpler terms, the function is decreasing at these points because the rate of change is negative.

3. How can I determine if a function is decreasing?

To determine if a function is decreasing, you can look at the slope of the function. If the slope is negative, then the function is decreasing. Another way is to look at the graph of the function and see if it is sloping downwards from left to right.

4. What is the significance of understanding why g is decreasing at x=2 and x=-2?

Understanding why g is decreasing at x=2 and x=-2 is important in analyzing and interpreting the behavior of the function. It can help in making predictions about the function's values at other points, as well as in identifying any potential maximum or minimum points.

5. Can a function be both increasing and decreasing at the same time?

No, a function cannot be both increasing and decreasing at the same time. A function can only have one type of behavior at a specific point. However, a function can have different behaviors at different points, such as increasing at some points and decreasing at others.

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