Find all points on the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x + 2 }$ where the tangent line is horizontal.
A) (0, 0) and (-4, -8)
B) There are none.
C) (0, 0)
D) (2, 1)

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 $f(x) = \frac{x^2}{x+2}$ is found using the quotient rule: $f'(x) = \frac{(2x)(x+2) - (x^2)(1)}{(x+2)^2} = \frac{2x^2 + 4x - x^2}{(x+2)^2} = \frac{x^2 + 4x}{(x+2)^2}$. Setting this equal to zero gives $x^2 + 4x = 0$, which factors to $x(x + 4) = 0$, yielding $x = 0$ and $x = -4$. Substituting these back into the original function gives the points (0, 0) and (-4, -8) where the tangent line is horizontal.

Find All Points on the Graph of the Function f(x)=x2x+2f ( x ) = \frac { x ^ { 2 } } { x + 2 }f(x)=x+2x2​

Find All Points on the Graph of the Function $f ( x ) = \frac { x ^ { 2 } } { x + 2 }$