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

To find points where the tangent line is horizontal, set $f'(x) = 0$: $-12x^3 + 12x = 0$. Factoring out $12x$, we get $12x(-x^2 + 1) = 0$, which gives $x = 0, \pm 1$. 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 f(x)=−3x4+6x2,f′(x)=−12x3+12xf ( x ) = - 3 x ^ { 4 } + 6 x ^ { 2 } , f ^ { \prime } ( x ) = - 12 x ^ { 3 } + 12 xf(x)=−3x4+6x2,f′(x)=−12x3+12x

Use the Function Below and Its Derivative to Determine Any $f ( x ) = - 3 x ^ { 4 } + 6 x ^ { 2 } , f ^ { \prime } ( x ) = - 12 x ^ { 3 } + 12 x$