1. ## A question about curve sketching and imaginary numbers

Hello.

I was wondering, what does it mean in terms of graphing when I bump into "i" during curve sketching? For example, in this particular "sketch the curve" question, in my check for extrema points I'm getting x to be an imaginary number:

first derivative check for maxs and mins:

or

Does this just mean no max or min exists?

2. Originally Posted by shawli
Hello.

I was wondering, what does it mean in terms of graphing when I bump into "i" during curve sketching? For example, in this particular "sketch the curve" question, in my check for extrema points I'm getting x to be an imaginary number:

first derivative check for maxs and mins:

or

Does this just mean no max or min exists?
Yes. Notice that the derivative is negative everywhere it's defined, so the function is decreasing everywhere.

3. Originally Posted by shawli
Hello.

I was wondering, what does it mean in terms of graphing when I bump into "i" during curve sketching? For example, in this particular "sketch the curve" question, in my check for extrema points I'm getting x to be an imaginary number:

first derivative check for maxs and mins:

or

Does this just mean no max or min exists?

Of course. The function is a real one so its derivative never vanishes within the real numbes and thus there are no extreme points. In fact, $f'(x)<0 \,\,\forall x \in \mathbb{R}$ so this is a monotonically descending function and, thus again, without extreme points.

Tonio