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B Do all the peaks and valleys of f have f'=0

  1. Apr 22, 2017 #1
    ebQ0uOS.png

    I learned in the earlier chapters that peaks and valleys of a fxn have points where f'=0 (i marked them with red x). A few chapters later it said if a fxn has 2 roots, then f'=0 (still the 1st graph).

    So does that mean if the graph of a fxn is like the 2nd graph, the peaks and valleys are not f'=0? I drew where i would assume f'=0 with blue circles
     
  2. jcsd
  3. Apr 22, 2017 #2
    If there is a knotch then f'(x) is not equal to zero......
     
  4. Apr 22, 2017 #3
    All peaks and all valleys have f'(x)=0 but there can be a point where f'(x)=0 but it's not peak or valley. For example f(x)=x^3 at x=0.

    The correct wording is that if a function has a double root, not two roots, then f'(x)=0 at that point. Happens for example with f(x)=x^2 at x=0.
     
  5. Apr 22, 2017 #4
    Take the case y=-|x| at x=0
     
  6. Apr 22, 2017 #5
    Yes the rules only apply if the function has a continuous derivative.
    With non-continuous functions or derivatives, pretty much anything can happen.
     
  7. Apr 26, 2017 #6
    What it should say, or maybe what you meant to say, was that if a function has two roots, say ##x_1## and ##x_2##, then there is a point ##c## between ##x_1## and ##x_2## such that ##f'(c) = 0##
     
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