Use the function below and its derivative to determine any points on the graph of $ f $ at which the tangent line is horizontal.
$f(x)=3 x^{4}-6 x^{2}, \quad f^{\prime}(x)=12 x^{3}-12 x$
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 $12x$, we get $12x(x^2 - 1) = 0$, which gives $x = 0, x = 1, x = -1$. Substituting these values into the original function $f(x) = 3x^4 - 6x^2$ gives the points $(-1, -3), (0, 0), (1, -3)$.

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

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