Characteristics of combined functions

Mar 2017
356
3
Massachusetts
Hi,

I hope someone can help. I'm hoping that someone help me understand combined function characteristics with the following examples:

1. y=2^x-x^3 would this function not have a relative maximum at x = 0.59? Would it also have a minimum at x = 8.177?
2. y = sin(2pix) - 2sin(pix) my textbook says that the zero/x-intercept for this function is k. Which doesn't make any sense to me. Can someone please tell me whether this is correct to think that the zero is k?
3. y = sin(2pix) + 2 + (1/x) my textbook says that this function has zeros that are changing at irregular intervals - however, I personally don't see where the function experiences any zeros in the first place. Am I understanding this correctly?
4. y = sin(2pix) + 2 + (1/x) The textbook is also saying for the same function as the one above that the maximum and minimum values are at irregular intervals. I don't see how this is possible. Do you think that my textbook is referring to local maximum or minimum values?

I know these are a lot of questions, but there wasn't a lot of information online that I could find to help me. Again, I really hope someone can help!

Sincerely,
Olivia
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
1. y=2^(x-x^3)
I assume you mean $y = 2^{x-x^3}$ ... if so, enclose the entire exponent in parentheses

relative max at $x \approx 0.577$, real min at $x \approx -0.577$


2. y = sin(2pix) - 2sin(pix)
$y = 2\sin(\pi x)\cos(\pi x) - 2\sin(\pi x) = 0$

$2\sin(\pi x)[\cos(\pi x) - 1= 0 \implies x = k, \, k \in \mathbb{Z}$


3 and 4. y = sin(2pix) + 2 + (1/x)
$y = 0$ at $x = -\dfrac{1}{2}$

yes, relative extrema
 
Mar 2017
356
3
Massachusetts
Thanks for everyone's help. I understand now what I was confused about.