Not a "normal" question

Calculus Level 3

Sketch the probability density function of a normal distribution with some mean and standard deviation. Call this graph \(f(x)\).

Define \(g(n)=\) number of turning points \(f^{(n)}(x)\) has, where \(f^{(n)}(x)\) is the \(n^{th}\) derivative of \(f(x)\).

What is the value of \(\sum_{n=0}^{20} g(n)\)?

BONUS: can you generalise this for \(\sum_{n=0}^{k} g(n)\)?


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