How many points of inflection does the function $f ( x ) = 3 x ^ { 5 } - 10 x ^ { 3 } + 5$ have?
A)5
B)0
C)2
D)7
E)6
F)1
G)3
H)4

Question

Quizplus · Accepted Answer

The function $f(x) = 3x^5 - 10x^3 + 5$ has points of inflection where its second derivative changes sign. The second derivative is $f''(x) = 60x^3 - 60x$. Setting $f''(x) = 0$ gives $x(x^2 - 1) = 0$, which has solutions $x = 0, 1, -1$. Checking intervals around these points shows that the sign of $f''(x)$ changes at each, indicating 3 points of inflection.

How Many Points of Inflection Does the Function f(x)=3x5−10x3+5f ( x ) = 3 x ^ { 5 } - 10 x ^ { 3 } + 5f(x)=3x5−10x3+5

How Many Points of Inflection Does the Function $f ( x ) = 3 x ^ { 5 } - 10 x ^ { 3 } + 5$