Bei Deepseek fällt mir auf, das es ellenlang laut vor sich hin denkt:
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Okay, let's see. I have to figure out if there are values of m (natural numbers greater than 2) for which the polynomial p(x) = x^m · (x² - 1) has a extremum or a inflection point at x = 0. Hmm, right. So first, I need to recall what conditions are required for extrema and inflection points.
For extrema, I remember that the first derivative should be zero, and the second derivative should be non-zero (positive for a minimum, negative for a maximum). For inflection points, the second derivative should be zero, and the third derivative should be non-zero. But wait, maybe there are other conditions too, especially if higher derivatives are also zero? Hmm, maybe I need to check the first non-zero derivative after the second for inflection points? Let me confirm that later.
First, let's compute the first derivative of p(x). The function is p(x) = x^m (x² - 1). Let's expand that first to make differentiation easier. So, expanding it, we get p(x) = x^{m+2} - x^m. Then, the first derivative p'(x) would be (m+2)x^{m+1} - m x^{m-1}.
Now, let's evaluate p'(0). If x = 0, then each term has a factor of x, except if the exponent is zero. But in p'(x), the exponents are m+1 and m-1. Since m is a natural number greater than 2, m ≥ 3. Therefore, m+1 ≥ 4 and m-1 ≥ 2. Thus, both terms have x raised to at least the power of 2. Therefore, substituting x = 0 into p'(x) gives 0. So, the first derivative at x = 0 is zero. That means x = 0 is a critical point.
Next, to determine if it's a maximum or minimum, we need the second derivative. Let's compute p''(x). Starting from p'(x) = (m+2)x^{m+1} - m x^{m-1}, the second derivative is p''(x) = (m+2)(m+1)x^m - m(m-1)x^{m-2}…….
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Und so geht das noch wirklich seitenweise (das waren etwa 10%) weiter bis dann endlich eine Antwort kommt. Wenn das kürzer und prägnanter wäre, könnte ich mir das als eine Art Tutor vorstellen. So ist das nur nervig und dauert auch sehr lange, währen ChatGPT und andere recht schnell zum Ergebnis kommen und man Erklärungen nur auf Nachfragen bekommt (wenn man richtig fragt).
Amüsant ist, wie oft triviale Fehler (alle geraden Zahlen wie 4, 5, 6, 8…) gemacht werden und auf entsprechende Hinweise beliebig oft, ‚sorry, wir machen das neu‘ kommt.