Determine any points on the graph of the following function at which the tangent line is horizontal.
$$f ( x ) = x ^ { 2 } + 6 x - 8$$
A) $( 3,19 )$
B) $( 3,0 )$
C) $( - 3,0 )$
D) $( 1 , - 1 )$
E) $( - 3 , - 17 )$

Question

Quizplus · Accepted Answer

To find points where the tangent line is horizontal, we need to find where the derivative of the function equals zero. The derivative of $f(x) = x^2 + 6x - 8$ is $f'(x) = 2x + 6$. Setting this equal to zero gives $2x + 6 = 0$, which simplifies to $x = -3$. Substituting $x = -3$ back into the original function gives $f(-3) = (-3)^2 + 6(-3) - 8 = 9 - 18 - 8 = -17$. Therefore, the point at which the tangent line is horizontal is $(-3, -17)$.

Determine Any Points on the Graph of the Following Function f(x)=x2+6x−8f ( x ) = x ^ { 2 } + 6 x - 8f(x)=x2+6x−8

Determine Any Points on the Graph of the Following Function $f ( x ) = x ^ { 2 } + 6 x - 8$