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

[UrbanXrisis]

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

 

[what]

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).

 

[UrbanXrisis]

no, it states between -2 and 2 AND when they are equal. why?

 

[UrbanXrisis]

$$-2 \underline{<}x\underline{<}2$$
is different from:
$$-2 < x < 2$$

why would the slope be decreasing at x=-2 and x=2 when the derivative is zero?

 

[what]

I misunderstood your question sorry about that.The first derivative test(straight from a calc book) states:

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

*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.