1. Identifying Inflection Points

The x coordinates of the points of inflection of the graph of $\displaystyle y= x^5-5x^4=3x+7$ are...

A 0,
B 1,
C 3,
D 0 and 3,
E 0 and 1

$\displaystyle y''=20x^3-60x^2$
Simplify.
$\displaystyle y''=20x^2(x-3)$

So, thanks to the graph and knowledge of the definition of inflection point (y''= 0).
I choose D.

However, the book says C.
I'm inclined to believe it is right because I did the sign test for 0 for numbers less than 0 and less than 3, thought I may be incorrect of which function to put the test values into (y'' right?), and got the same sign (negative).

So, did I do the sign test wrong? Or is the right and I'm confusing myself with these questions.

What is the answer to the problem?

Thank you.

2. For a point of inflection, $\displaystyle y''=0$ is a necessary condition but not a sufficient one. $\displaystyle y''$ must also change sign before and after the point of inflection. Thus 0 is not a point of inflection because $\displaystyle y''$ does not change sign there; $\displaystyle y''$ is negative both slightly before and slightly after 0.

Alternatively …

$\displaystyle y'''=60x^2-120x\ne0$ when $\displaystyle x=3$. Since $\displaystyle y'''$ is an odd-order derivative, $\displaystyle x=3$ is a point of inflection. However $\displaystyle y'''(0)=0$ so we differentiate again.

$\displaystyle y''''=120x-120\ne0$ when $\displaystyle x=0$. $\displaystyle y''''$ is an even-order derivative, so $\displaystyle x=0$ is not a point of inflection.