Is it an inflection point if the second derivative is a double root? For example, would the function $f''(x)=x^2(x-3)(x-6)$ have 3 inflection points or 2? Although it is clear that 3 and 6 are inflection points, would 0 be a point of inflection or a bounce?
Is it an inflection point if the second derivative is a double root?
0
$\begingroup$
calculus
-
0what is your $f(x)$? – 2017-01-20
1 Answers
3
When we are unsure, we should refer to the definitions.
An inflection point is a point where the curvature changes its sign while a tangent exists. In this case, the curvature doesn't change sign at $0$, hence it is not an inflection point.
-
0How would you know where the curvature changes signs? Is this because a square is always positive? – 2017-01-20
-
1For a polynomial, we can check its second derivative. If the power of the root of the second derivative is odd, then it changes sign. – 2017-01-20
-
1@DiscreteMath Essentially yes. The other two factors don't change signs crossing $x=0$, and although $x$ does, $x^2$ does not. – 2017-01-20