Use the function and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.Use a graphing utility to verify your results. $f ( x ) = 5 x ^ { 6 } + 6 x ^ { 5 } , \quad f ^ { \prime } ( x ) = 30 x ^ { 5 } + 30 x ^ { 4 }$
A) $f$ has vertical tangents at $( - 1 , - 1 )$ and $( 0,0 )$ .
B) $f$ has horizontal tangents at $( 1,1 )$ and $( 0,0 )$ .
C) $f$ has horizontal tangents at $( - 1 , - 1 )$ and $( 0,0 )$ .
D) $f$ has vertical tangents at $( 1,1 )$ and $( 0,0 )$ .
E) $f$ has vertical tangents at $( - 1 , - 1 )$ and $( 1,1 )$ .

Question

Use the function and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.Use a graphing utility to verify your results.   $f ( x ) = 5 x ^ { 6 } + 6 x ^ { 5 } , \quad f ^ { \prime } ( x ) = 30 x ^ { 5 } + 30 x ^ { 4 }$  
A) $f$ has vertical tangents at $( - 1 , - 1 )$ and $( 0,0 )$ .
B) $f$ has horizontal tangents at $( 1,1 )$ and $( 0,0 )$ .
C) $f$ has horizontal tangents at $( - 1 , - 1 )$ and $( 0,0 )$ .
D) $f$ has vertical tangents at $( 1,1 )$ and $( 0,0 )$ .
E) $f$ has vertical tangents at $( - 1 , - 1 )$ and $( 1,1 )$ .

Quizplus · Accepted Answer

Horizontal tangents occur where the derivative $f'(x) = 30x^5 + 30x^4$ equals zero. Factoring out $30x^4$ gives $30x^4(x + 1) = 0$, which has solutions at $x = 0$ and $x = -1$. The points on $f(x)$ at these $x$ values are $(0,0)$ and $(-1, -1)$, respectively, indicating horizontal tangents at these points.

Use the Function and Its Derivative to Determine Any Points fff

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