frx 2 years ago 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. frx

2. 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 ....

3. frx

There are multiple x values then, x=-2.5; x=-1.5; x=-0.5; x=0.5; x=1.5;x=2.5

4. frx

in the interval -3 to 3

5. A_clan

So, you will have a range of values and not just one value in the solution

6. A_clan

x={-2.5,-1.5,-0.5,0.5,1.5,2.5}

7. frx

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?

8. A_clan

It is possible to see it

9. A_clan

Is there any point on the curve where , on drawing a straight line, you get a horizontal line ?

10. frx

I can't see it, no. The x-axis itself?

11. A_clan

|dw:1365268526951:dw| This is one example

12. frx

Are there two horizontal tangents then? Going through the points of the curvatures?

13. A_clan

No, every point of curvature has its own tangent

14. frx

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?

15. A_clan

My experience is, In mathematics, the grammar of the question may not always indicate the quantity of solution.

16. frx

So the solutionset is {-3,-2,-1,0,1,2,3} then ?

17. A_clan

yes, that looks right

18. frx

Great! What about the local maximum and minimum, there seems to be muliple max and mins too?

19. A_clan

The solution to last part is situational. Local maximum and minimum may change depending upon the interval you choose for calculating them.

20. frx

The interval is -3 to 3, like the graph shows

21. 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.

22. 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'

23. A_clan

|dw:1365269863585:dw|

24. A_clan

A,B and C are the local maxima but out of these, B is the global maxima

25. frx

Sure i understand , so the local maximas are at -3, -1, 1 and 3 and the local mins are -2,0,2

26. A_clan

That seems right

27. frx

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

28. 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 .

29. A_clan

Then correct solution should be -1 and 1

30. frx

Ok, that makes sense :)

31. frx

Thanks you so much for your help, really appreciate it!

32. A_clan

no problem