We are given the curve $y = 1 + 3x^{\frac{2}{3}} + x^2$, and we need to find where the tangents at the points where $y = 1$ intersect.

Step 1: Find the points where $y = 1$

To find the points where $y = 1$, we need to solve the equation: $1 + 3x^{\frac{2}{3}} + x^2 = 1$ Simplifying: $3x^{\frac{2}{3}} + x^2 = 0$ Factor the equation: $x^{\frac{2}{3}}(3 + x^{\frac{4}{3}}) = 0$ This gives two solutions:

1. $x^{\frac{2}{3}} = 0$ ⇒ $x = 0$
2. $3 + x^{\frac{4}{3}} = 0$ ⇒ $x^{\frac{4}{3}} = -3$, but this has no real solutions because $x^{\frac{4}{3}}$ cannot be negative for real $x$.

Thus, the only solution is $x = 0$.

Step 2: Find the equation of the tangent line at $x = 0$

To find the tangent line at $x = 0$, we need to calculate the derivative of the curve $y = 1 + 3x^{\frac{2}{3}} + x^2$.

The derivative of the function is: $\frac{dy}{dx} = \frac{d}{dx}\left( 1 + 3x^{\frac{2}{3}} + x^2 \right)$ Using the power rule for each term: $\frac{dy}{dx} = 0 + 3 \cdot \frac{2}{3}x^{-\frac{1}{3}} + 2x = 2x^{-\frac{1}{3}} + 2x$ At $x = 0$, the first term $x^{-\frac{1}{3}}$ is undefined, so the slope of the tangent line at this point does not exist. This means the tangent line is vertical at $x = 0$.

Step 3: Check for other points of intersection

Since the only real point where $y = 1$ is $x = 0$, and the tangent line at $x = 0$ is vertical, there is no other point where the tangent line intersects the curve.

Thus, the tangent to the curve at the point where $y = 1$ is a vertical line at $x = 0$. It does not intersect the curve at any other point.

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Would you like more details or have any further questions?

Related Questions:

1. How do you find the equation of a tangent line at a specific point?
2. What happens when a tangent line is vertical?
3. Can you explain how to differentiate fractional powers like $x^{\frac{2}{3}}$?
4. How do vertical tangents affect the graph of a function?
5. Why doesn't $x^{\frac{4}{3}} = -3$ have real solutions?

Tip:

When differentiating fractional powers, remember that negative exponents can lead to undefined points, especially at $x = 0$.