the tangent to the curve y=1+3x^2/3+x^2 draw at the points for which y=1, intersect at which point

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 \( x = 0 \).

The tangent to the curve y = 1 + 3x^(2/3) + x^2 drawn at the points for which y = 1 intersects at which point?

Discover how to find the tangent to the curve y = 1 + 3x^(2/3) + x^2 at points where y = 1 and determine where these tangents intersect. This problem involves concepts such as calculus, derivatives, tangents, and curve sketching, making it suitable for students in Grades 11-12.

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