Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.
$$f ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } , f ^ { \prime } ( x ) = 12 x ^ { 3 } - 12 x$$
A) $( - 1 , - 3 ) , ( 0,0 ) , ( 1 , - 3 )$
B) $( - 1,3 ) , ( 1,12 )$
C) $( - 1 , - 3 ) , ( 1 , - 3 )$
D) $( 0,0 ) , ( 1,0 )$
E) $( 0,0 ) , ( 1 , - 3 )$

Question

Quizplus · Accepted Answer

The tangent line is horizontal where the derivative $f'(x) = 12x^3 - 12x = 0$. Factoring out 12x gives $x( x^2 - 1 ) = 0$, so $x = 0, \pm 1$. Substituting these values into $f(x)$ gives the points $(-1, -3)$, $(0, 0)$, and $(1, -3)$.

Use the Function Below and Its Derivative to Determine Any fff

Use the Function Below and Its Derivative to Determine Any $f$