A graphing celculator is recommended.
(a) The curve with ecuation is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point ( 1,2 )
(t) At what points does this curve have horizontal tangents? (Order your answers from smallest to largest , then from smallest to largest )
( (b) by graphing the curve and the tangent lines on a common screen.

Ask by Simpson Fuentes. in the United States

Mar 20,2025

Question

A graphing celculator is recommended.
(a) The curve with ecuation is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point ( 1,2 )
(t) At what points does this curve have horizontal tangents? (Order your answers from smallest to largest , then from smallest to largest )
( (b) by graphing the curve and the tangent lines on a common screen.

Ask by Simpson Fuentes. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

It seems that the system does not currently support plotting the graphs directly. However, I can guide you on how to graph the curve and the tangent line using graphing software or a graphing calculator.

### Instructions for Graphing

1. **Graph the Curve**:
   - Use the equation $$ y^2 = x^3 + 3x^2 $$.
   - You can rearrange it to $$ y = \pm \sqrt{x^3 + 3x^2} $$ to plot both the positive and negative branches of the curve.

2. **Graph the Tangent Line**:
   - Use the equation $$ y = \frac{9}{4}x - \frac{1}{4} $$.
   - This is a straight line with a slope of $$ \frac{9}{4} $$ and a y-intercept of $$ -\frac{1}{4} $$.

3. **Set the Viewing Window**:
   - For both graphs, set the x-axis from approximately -4 to 2 and the y-axis from -3 to 3 to capture the relevant features of the curve and the tangent line.

### Points to Mark
- Mark the point of tangency (1, 2) on the graph.
- Also, mark the points where the curve has horizontal tangents: $$ (-2, -2) $$, $$ (-2, 2) $$, and $$ (0, 0) $$.

### Conclusion
By following these instructions, you should be able to visualize the Tschirnhausen cubic and its tangent line effectively. If you have access to graphing software like Desmos, GeoGebra, or a graphing calculator, you can input the equations directly to see the graphs.
To find the tangent line at (1,2), the equation is $$ y = \frac{9}{4}x - \frac{1}{4} $$. The curve has horizontal tangents at $$ (-2, -2) $$, $$ (-2, 2) $$, and $$ (0, 0) $$.

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