How are critical points related to local and absolute extrema?

1 Answer

Critical points are locations on a function graph where the derivative is equal to zero or doesn't exist.


#f'(x)=0# is the notation you might see for when the derivative equals 0. The graph might have a nice "bump" or "dip" where a tangent line would be horizontal.

My screenshot

This parabola has an absolute maximum at the critical point since (1.5, -1.75) is the vertex. The tangent line at this point has a slope of zero. The function increases toward the vertex from the left, and then decreases away from the vertex on the right.

For some functions, a critical point will occur at a change in concavity. In the next function, the critical point is at (4,2), where the curve goes from concave up to concave down. This point does not represent a max or min, even though the slope of the tangent line is zero. In order to be a max or min (absolute or relative), the function must either increase and then decrease or vice versa. On both sides of this critical point, the function continues to increase from left to right. No max or min. We would probably call this an inflection point. (further tests are needed to determine...) [Univ of Chicago]

My screenshot 2

For some functions, there is no derivative at a certain point. An absolute value graph comes to a point at the tip of the "v" shape, or vertex. This point would be a critical point, and an absolute max or min, but the derivative there does not exist. derivative of absolute value

My screenshot 3

But wait, there's more! What about a function like y = |sin(x)|? This function has some nice "bumps" (relative max) but also some cusps!
Not differentiable at certain values of x

As you can see from the graph, there are many locations that will provide a maximum value of 1, but also many other locations where you see a cusp. These critical points occur at odd integer multiples of #pi/2#, whereas the minimum values of 0 occur at even integer multiples of #pi/2#.

My screenshot 4