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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.
(1) By definition, An infinite discontinuity occurs when there is a vertical asymptote at the given x value. So, your x value will be ....
There are multiple x values then, x=-2.5; x=-1.5; x=-0.5; x=0.5; x=1.5;x=2.5
So, you will have a range of values and not just one value in the solution
x={-2.5,-1.5,-0.5,0.5,1.5,2.5}
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?
It is possible to see it
Is there any point on the curve where , on drawing a straight line, you get a horizontal line ?
I can't see it, no. The x-axis itself?
|dw:1365268526951:dw| This is one example
Are there two horizontal tangents then? Going through the points of the curvatures?
No, every point of curvature has its own tangent
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?
My experience is, In mathematics, the grammar of the question may not always indicate the quantity of solution.
So the solutionset is {-3,-2,-1,0,1,2,3} then ?
Great! What about the local maximum and minimum, there seems to be muliple max and mins too?
The solution to last part is situational. Local maximum and minimum may change depending upon the interval you choose for calculating them.
The interval is -3 to 3, like the graph shows
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.
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,B and C are the local maxima but out of these, B is the global maxima
Sure i understand , so the local maximas are at -3, -1, 1 and 3 and the local mins are -2,0,2
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
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 .
Then correct solution should be -1 and 1
Thanks you so much for your help, really appreciate it!