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Algebra Level 4

Find the sum of all numbers n n such that there can be n n points of inflection on the graph of a polynomial of degree 7.


The answer is 9.

Otto Bretscher by Otto Bretscher , 65, USA

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1 solution

Samuel Jones
May 13, 2015

There are as many points of inflection as the number of real roots of the double derivative of the polynomial. Thus, since the polynomial is 7 t h 7^{th} degree, the double derivative must be 5 t h 5^{th} degree. And, since complex roots occour in conjugate pairs for a polynomial with real coefficients, we have,

either 1 , 3 , or 5 real roots \text{either} \ 1, \ 3, \ \text{or} \ 5 \ \text{real roots} \

⟹ ∑ n = 9 \displaystyle \implies \sum {n} = \boxed{9}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes, your answer is correct, but you need to think about multiple roots of the second derivative. It's not quite correct to say that "There are as many points of inflection as the number of real roots of the double derivative of the polynomial".

Otto Bretscher - 6 years, 1 month ago

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