anonymous
  • anonymous
I have the function \[f(x)=\frac{ \sin(x \pi ) }{ \sin(2x \pi) }\] which is visually presented in the attached file. I'm supposed, only by looking at the graph, to decide * Where in the interval the function is discontinuous? * Where does the curve have a horizontal tangent? * Where does the function have local maximum, local minimum? I can't see these things and would hence appreciate some guidance.
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
1 Attachment
A_clan
  • A_clan
(1) By definition, An infinite discontinuity occurs when there is a vertical asymptote at the given x value. So, your x value will be ....
anonymous
  • anonymous
There are multiple x values then, x=-2.5; x=-1.5; x=-0.5; x=0.5; x=1.5;x=2.5

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anonymous
  • anonymous
in the interval -3 to 3
A_clan
  • A_clan
So, you will have a range of values and not just one value in the solution
A_clan
  • A_clan
x={-2.5,-1.5,-0.5,0.5,1.5,2.5}
anonymous
  • anonymous
I get it, thanks! Then how about a horizontal tangent, don't I have to calculate the lim as f(x) goes to infinity or is it possible to see it?
A_clan
  • A_clan
It is possible to see it
A_clan
  • A_clan
Is there any point on the curve where , on drawing a straight line, you get a horizontal line ?
anonymous
  • anonymous
I can't see it, no. The x-axis itself?
A_clan
  • A_clan
|dw:1365268526951:dw| This is one example
anonymous
  • anonymous
Are there two horizontal tangents then? Going through the points of the curvatures?
A_clan
  • A_clan
No, every point of curvature has its own tangent
anonymous
  • anonymous
So there are seven horizontal lines which are tangents but the question is asked in singular which should indicate that there where only one answer?
A_clan
  • A_clan
My experience is, In mathematics, the grammar of the question may not always indicate the quantity of solution.
anonymous
  • anonymous
So the solutionset is {-3,-2,-1,0,1,2,3} then ?
A_clan
  • A_clan
yes, that looks right
anonymous
  • anonymous
Great! What about the local maximum and minimum, there seems to be muliple max and mins too?
A_clan
  • A_clan
The solution to last part is situational. Local maximum and minimum may change depending upon the interval you choose for calculating them.
anonymous
  • anonymous
The interval is -3 to 3, like the graph shows
A_clan
  • A_clan
We say 'a' is local maximum if the height of the function at 'a' is greater than (or equal to) the height anywhere else in that interval.
A_clan
  • A_clan
In simple words, we can say, local maxima is the maximum height in certain part of the graph but if we consider the whole graph ,one of these has to be the 'global maximum'
A_clan
  • A_clan
|dw:1365269863585:dw|
A_clan
  • A_clan
A,B and C are the local maxima but out of these, B is the global maxima
anonymous
  • anonymous
Sure i understand , so the local maximas are at -3, -1, 1 and 3 and the local mins are -2,0,2
A_clan
  • A_clan
That seems right
anonymous
  • anonymous
Should I include the endpoints -3 and 3 according to WA they are not included http://www.wolframalpha.com/input/?i=Local+max+sin%28x+pi%29%2Fsin%282x+pi%29+from+-3+to+3
A_clan
  • A_clan
It seems the local maxima and local minima are left out at the boundary because we do not have the values of y from both left side and right side,when x is approaching -3 or 3 .
A_clan
  • A_clan
Then correct solution should be -1 and 1
anonymous
  • anonymous
Ok, that makes sense :)
anonymous
  • anonymous
Thanks you so much for your help, really appreciate it!
A_clan
  • A_clan
no problem

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