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Finding critical point

  1. Oct 21, 2006 #1
    Ok so I am trying to do this problem and I have a question

    So based on the definition given in the book "An interior point of the domain of a function f where f' is zero or undefined is a critical point of f"

    This is the problem:
    y = sqrt(x^2 - 1)
    y' = x/sqrt(x^2 - 1)

    to find a critical point
    y' = 0
    x/sqrt(x^2 - 1) = 0
    x = 0

    also to find the critical point we have to see if y' will be undefined at any value of x. as we can see y' will be undefined at x = 0.

    so from the first condition when we solved for y' = 0, we got x = 0 and now for the second condition y' is undefined at x = 0.

    So both conditions are satisfied at x = 0 so does that mean the critical point is at x = 0. In the definition it says first condition or second condition has to be satisfied. I might be reading too much into the definition.
  2. jcsd
  3. Oct 24, 2006 #2
  4. Oct 24, 2006 #3
    if x = -1, or 1 then f' is undefined. 1 and -1 also exist in f(x).

    -1 < x < 1 do not exist (real) in f(x) so they are not critical points
    when x = 0 (included in the inequality above) f(x) does not exist.

    Notice f(x=0) means, sqrt(0^2-1) = sqrt(-1) = i
  5. Oct 25, 2006 #4


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    What IS the domain of that function?
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