(3.) Extrempunkte bestimmen, notwendiges und hinreichendes Kriterium

In this video, you'll learn how to determine extrema (maximum and minimum points) and what the necessary and sufficient criteria are all about. Maximum and minimum points are local/relative extrema. Tangents passing through local maxima and minimum points have a horizontal tangent. Therefore, to determine extrema, you first look for the zeros of the first derivative. Because the slope is zero at all relative extrema, this approach finds all possible relative extrema (xE). Therefore, f'(xE) = 0 is the necessary condition for the existence of an extremum. But be careful: Saddle points are not extrema, but they also have a horizontal tangent and fulfill the necessary condition for extrema. That's why a second criterion—the sufficient criterion—is needed to distinguish between "true" relative extrema and saddle points. In this video, you'll see that there are two different types of sufficient criteria: (a) a sign change at the zero of the first derivative, and (b) that the second derivative f''(xE) is not equal to zero at the candidate extremum. This video introduces the theory; the next video will demonstrate concrete examples. This video is part of a series on function analysis using derivatives; the playlist is available here:    • Funktionsuntersuchung mittels Ableitungen   See also: ** See the Differential Calculus playlist for all videos on differential calculus:    • Differentialrechnung   🧑‍🎓Need tutoring? - No problem - Book a free, no-obligation initial consultation: https://tidycal.com/mathehoch13/15-mi... Call ID: m13v0274 ** Subscribe to my channel here so you never miss a video: http://www.youtube.com/user/MaNHinDo?... ** Facebook page of mathehoch13. You can also leave comments, give feedback, or suggest video topics here:   / mathehoch13   ** If my videos have helped you, I'm always happy to receive: :) a "Like" 8) if you share my videos with classmates and friends :D comments with questions, suggestions, or video requests :P if you subscribe to my channel. All the best and see you next time, Your Math Coach, Christoph Goemans