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Graph of $f'(x)$

enter image description here

Supposedly there is no local minimum on the interval (0, 8) for the above graph, but I don't understand why. I thought there were local mins at x = 1 and x = 6.

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  • $\begingroup$ The function being graphed has local minima, but your link suggests that this is the graph of the derivative of the function you are inspecting? $\endgroup$ – turkeyhundt Dec 15 '16 at 21:14
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    $\begingroup$ You have the graph of $f'(x)$, while I suspect that the question asks about the minima of $f(x)$. Can that be right? $\endgroup$ – Arthur Dec 15 '16 at 21:14
  • $\begingroup$ Oh! It is asking for the minima of f(x). Is the minima of f(x) where f'(x) is above 0? $\endgroup$ – user389369 Dec 15 '16 at 21:59
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It sounds like the exercise is asking you to explain that there are no local minima of $f$, providing you with the graph of $f'$. If this interpretation of the exercise is correct, consider that candidates for local extrema of $f$ are at zeros of $f'$. There is one zero of $f'$ in the middle of the picture, but no sign change in $f'$ occurs there, so this is not a local extremum. (Near this point the graph would look something like the graph of $y=-x^3$ near $x=0$.)

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  • $\begingroup$ Why the downvote? $\endgroup$ – Ian Dec 15 '16 at 23:02
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Notice that this is the graph of the derivative of f(x). Does your teacher mean to find local max/mins for the original function? If so, f(x) would have no local min because notice the slope (f'(x)) is always zero except at x=4 which indicates an inflection point.

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    $\begingroup$ Not of the function $f$. $\endgroup$ – Brian M. Scott Dec 15 '16 at 21:14
  • $\begingroup$ The link to the graph says "Graph of f'(x)". So, yeah, it was stated in the question, Tye. $\endgroup$ – amWhy Dec 15 '16 at 21:20
  • $\begingroup$ Since the OP has been told that whatever function is under consideration has no local minimum in $(0,8)$, and since we’re clearly given the graph of $f'$, not of $f$, it is easily inferred from the available information that the original question was almost certainly about $f$, not about $f'$. $\endgroup$ – Brian M. Scott Dec 15 '16 at 21:21
  • $\begingroup$ @BrianM.Scott you are correct. I did not notice the ' on the link f'(x). Thank you. $\endgroup$ – MathGuy Dec 15 '16 at 21:24
  • $\begingroup$ You’re welcome; sorry if I sounded a bit abrupt. $\endgroup$ – Brian M. Scott Dec 15 '16 at 21:26

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