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Piecewise function

  • Thread starter Kolika28
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  • #1
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Homework Statement:

##f(x)=\left\{
\begin{array}{ll}
\frac{e^{-x}-1}{x}, & x>0 \\
\frac{x}{2}+1, & x\leq 0 \\
\end{array}
\right.##

a) At which intervals are f strictly increasing and at what intervals are f strictly decreasing.
b) Determine any local and global extreme values for f.
c) Determines if f has asymptotes.

Relevant Equations:

The derivative
a) At which intervals are f strictly increasing and at what intervals are f strictly decreasing.

Should I just find the derivative of both of the functions? If so, I get that the function is increasing at the intervals (−∞,0) and (0,∞). Is this right, or can I just say that the function is increasing at the interval (−∞,∞)?

b) Determine any local and global extreme values for f.

When graphing the function I don't see any local or global extreme values. f(x) consists of a straight line and curve with where f(x)=0 is not true. The whole function is not bounded, so I can't look at the values in the endpoints. But my teacher says there are extreme values. But how so?

c) Determines if f has asymptotes.

I know there is at least one horisontal asymptote, y=0, given the first function. Because
1572131368682.png

I was told there is one oblique asymptote also. But how?
 
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Answers and Replies

  • #2
Orodruin
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a) You would just say that the function is strictly increasing everywhere.

b) The function is bounded from above, but it has no extreme points. (It has a supremum, but no maximal value.)

c) what happens for large negative x?
 
  • #3
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a) You would just say that the function is strictly increasing everywhere.

b) The function is bounded from above, but it has no extreme points. (It has a supremum, but no maximal value.)

c) what happens for large negative x?
b) Should I just say that it has a supremum then in x=0??
c) For large negative x I look at ##\frac{x}{2}+1##. And I just get ##-∞## . But that does not tell me what the oblique slope is?
 
  • #4
LCKurtz
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a) You would just say that the function is strictly increasing everywhere.
Not on the interval [-1,1].

b) The function is bounded from above, but it has no extreme points. (It has a supremum, but no maximal value.)
##f(0) = 1## is an absolute maximum.
 
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  • #5
Orodruin
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Not on the interval [-1,1].



##f(0) = 1## is an absolute maximum.
Oops. I just assumed OP had checked for continuity at x=0...

b) Should I just say that it has a supremum then in x=0??
No, it has a global max in x=0.

c) For large negative x I look at x2+1x2+1\frac{x}{2}+1. And I just get −∞−∞-∞ . But that does not tell me what the oblique slope is?
You are not looking for a value of the function. You are looking for a line on the form ##kx+m## that the function approaches.
 
  • #6
WWGD
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b) Should I just say that it has a supremum then in x=0??
c) For large negative x I look at ##\frac{x}{2}+1##. And I just get ##-∞## . But that does not tell me what the oblique slope is?
The slope of a line is constant, here =1/2.
 
  • #7
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Ok, now I understand what you mean by task b. But I'm still confussed about c). So I'm supposed to find a line on the form ##kx+m##.
You are not looking for a value of the function. You are looking for a line on the form kx+mkx+mkx+m that the function approaches.
The slope of a line is constant, here =1/2.
So is the oblique tanget just ##\frac{x}{2}+1##?
 
  • #8
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Not on the interval [-1,1].
Hmm, why not this interval. Is the function decreasing here?
 
  • #9
Orodruin
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Hmm, why not this interval. Is the function decreasing here?
Not in the entire interval, no. You have (correctly) concluded that the function is increasing on (-infinity,0) and (0,infty). What is left to check?
 
  • #10
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Not in the entire interval, no. You have (correctly) concluded that the function is increasing on (-infinity,0) and (0,infty). What is left to check?
I'm sorry, but I'm really blank right now. I don't see what's left to check. Sorry, for all the questions by the way, I just really want to understand!
 
  • #11
Orodruin
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What is the value of f(0)? What is the value of f(0.000000001)?
 
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  • #12
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What is the value of f(0)? What is the value of f(0.000000001)?
Ohh, I understand now. Thank you so much!!!My last question is:
So is the oblique tanget just x2+1x2+1\frac{x}{2}+1?
 

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