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So I learned that in order to find the minimum/ maximum point of a function, I would take it's derivative and find critical points, points where the derivative is zero or undefined, and then put them on a number line and do strawberry field. If it's negative to the left of the point and positive to the right of the point, the point is a minimum, etc. What I'm confused about is when the derivative is undefined, or when the denominator of your fraction is equal to zero, isn't that where you have a vertical tangent line and not a candidate for a min/max point?

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    $\begingroup$ What does the term "strawberry field" refer to? The derivative might be undefined where the function goes to infinity, in which case find out whether positive or negative from each side. $\endgroup$ – Joffan Jul 4 '18 at 17:31
  • $\begingroup$ I think it's best to do these kinds of problems by sketching the graph and thinking about the picture rather than by trying to apply rules. Then you'll see the vertical tangents and corners and cusps. Corners and cusps might be maximum or minumum values. $\endgroup$ – Ethan Bolker Jul 7 '18 at 0:40
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isn't that where you have a vertical tangent line and not a candidate for a min/max point?

Not necessarily.

  • when the derivative is undefined

    $f(x)=|x|$ is not differentiable at $x=0$, yet $x=0$ is a global minimum.

  • or when the denominator of your fraction is equal to zero

    $f(x)=\sqrt{|x|}$ has a vertical tangent line at $x=0$, yet $x=0$ is a global minimum.

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You can apply this (first derivative test) only on intervals where the function is differentiable.

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