Horizontal tangent point on implicit curve (single-step)
A concise question that asks for the coordinates or x-coordinate of a point where the tangent to an implicitly defined curve is horizontal (dy/dx = 0), without requiring the student to first derive dy/dx as a separate show-that step.
Consider the part of the graph of $y ^ { 2 } + x ^ { 3 } = 15 x y$ that is strictly to the right of the $Y$-axis, i.e., take only the points on the graph with $x > 0$. Questions (33) Write the least possible value of $y$ among considered points. If there is no such real number, write NONE. (34) Write the largest possible value of $y$ among considered points. If there is no such real number, write NONE.
In the analytic plane $$x y ^ { 2 } - x ^ { 3 } y - 6 = 0$$ Given that the tangent line passing through the point $\mathbf { P } \left( \mathbf { x } _ { 0 } , \mathbf { y } _ { 0 } \right)$ on the curve given by the equation is parallel to the x-axis, what is $\mathrm { x } _ { 0 }$? A) $- 3$ B) $- 2$ C) $\frac { - 3 } { 2 }$