The tangent to the curve `y=xe^(x^2)` passing through the point (1,e) also passes through the point

Options

A. `(4/3,2e)`
B. (2, 3e)
C. `(5/3,2e)`
D. (3, 6e)

Answer

To solve the problem, we need to find the tangent to the curve \( y = x e^{x^2} \) that passes through the point \( (1, e) \) and then determine which of the given points it also passes through.

### Step 1: Find the derivative of the curve

The first step is to differentiate the function \( y = x e^{x^2} \) using the product rule.

\[ \frac{dy}{dx} = e^{x^2} + x \cdot \frac{d}{dx}(e^{x^2}) \]
Using the chain rule for \( e^{x^2} \):
\[ \frac{d}{dx}(e^{x^2}) = e^{x^2} \cdot 2x \]
Thus,
\[ \frac{dy}{dx} = e^{x^2} + x \cdot (e^{x^2} \cdot 2x) = e^{x^2}(1 + 2x^2) \]

### Step 2: Evaluate the derivative at the point \( (1, e) \)

Now, we need to find the slope of the tangent line at \( x = 1 \):
\[ \frac{dy}{dx} \bigg|_{x=1} = e^{1^2}(1 + 2 \cdot 1^2) = e(1 + 2) = 3e \]

### Step 3: Write the equation of the tangent line

The equation of the tangent line can be expressed in the point-slope form:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (1, e) \) and \( m = 3e \):
\[ y - e = 3e(x - 1) \]
This simplifies to:
\[ y = 3ex - 2e \]

### Step 4: Check which points satisfy the tangent line equation

Now we need to check which of the given points satisfies the equation \( y = 3ex - 2e \).

#### Check Point 1: \( \left( \frac{4}{3}, 2e \right) \)
Substituting \( x = \frac{4}{3} \):
\[ y = 3e \cdot \frac{4}{3} - 2e = 4e - 2e = 2e \]
This point satisfies the equation.

#### Check Point 2: \( (2, 3e) \)
Substituting \( x = 2 \):
\[ y = 3e \cdot 2 - 2e = 6e - 2e = 4e \]
This point does not satisfy the equation.

#### Check Point 3: \( \left( \frac{5}{3}, 2e \right) \)
Substituting \( x = \frac{5}{3} \):
\[ y = 3e \cdot \frac{5}{3} - 2e = 5e - 2e = 3e \]
This point does not satisfy the equation.

#### Check Point 4: \( (3, 6e) \)
Substituting \( x = 3 \):
\[ y = 3e \cdot 3 - 2e = 9e - 2e = 7e \]
This point does not satisfy the equation.

### Conclusion

The only point that the tangent line passes through, besides \( (1, e) \), is \( \left( \frac{4}{3}, 2e \right) \).

### Final Answer
The tangent to the curve \( y = x e^{x^2} \) passing through the point \( (1, e) \) also passes through the point \( \left( \frac{4}{3}, 2e \right) \).

---