Critical Points II

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Last time we looked at some critical points of functions where the derivative of the function was zero. Recall that there is another type of critical point: if f'(c) doesn’t exist but c is in the domain of f, then c is also a critical point.

How can this happen? Consider the function f(x)=|x|. This function has a “corner” when x=0, so f'(0) doesn’t exist. In fact, the derivative of f is pretty simple:

f'(x) = \begin{cases} 1 & \text{if } x > 0,\\ -1 & \text{if } x < 0. \end{cases}

Since zero isn’t in the domain of f', it is a critical point of f.

For another example, consider the function g(x)=\sqrt{|x|}. As x approaches zero, the slope of the tangent line to g approaches infinity or minus infinity. Therefore zero is not in the domain of g', and since zero is in the domain of g, zero is a critical point of g.

Notice that both of these functions have minimum values at the critical points we looked at.

Here are a couple of exercises to help you understand these kinds of critical points:

  • Calculate the derivative of f. Do you get the same expression I did? Why doesn’t f' exist at zero?
  • Graph g(x). What does it look like? Is it continuous? Is it “smooth?” Why doesn’t g' exist at zero?