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 to the graph of the function is horizontal, we need to find where the derivative of the function equals zero, as this indicates a slope of 0. 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 to find the corresponding y-coordinate 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 $f ( x ) = x ^ { 2 } + 6 x - 8$