Find the points at which the graph of the equation has a vertical or horizontal tangent line.
A)There is a vertical tangent at but no horizontal tangents.
B)There is a horizontal tangent at and a vertical tangent at .
C)There is a horizontal tangent at but no vertical tangents.
D)There is a horizontal tangent at and a vertical tangent at .
E)There are no horizontal or vertical tangent lines.

Question

Find the points at which the graph of the equation has a vertical or horizontal tangent line. ​  ​
A)There is a vertical tangent at but no horizontal tangents. 
B)There is a horizontal tangent at and a vertical tangent at . 
C)There is a horizontal tangent at but no vertical tangents. 
D)There is a horizontal tangent at and a vertical tangent at . 
E)There are no horizontal or vertical tangent lines.

Quizplus · Accepted Answer

To find horizontal tangent lines, we need to find points where the derivative of the equation is equal to 0. Taking the derivative of the equation, we get: 
$\frac{dy}{dx}=-\frac{4x}{(x^2+4)^2}$
Setting the derivative equal to 0, we get:
$-\frac{4x}{(x^2+4)^2}=0$
This is only true when $x=0$. Plugging this value into the equation gives the point $(0,1)$, where there is a horizontal tangent line.

To find vertical tangent lines, we need to find points where the derivative of the equation is undefined (i.e. where the denominator is equal to 0). Setting the denominator equal to 0, we get:
$x^2+4=0$
This has no real solutions, so there are no vertical tangent lines. Therefore, choice B is the best option.

Find the Points at Which the Graph of the Equation