The equation of tangent to the curve `y=x+(4)/(x^2)`, that is parallel to the x-axis is y = k. Then the value of k is?

Answer

To find the value of \( k \) for the tangent to the curve \( y = x + \frac{4}{x^2} \) that is parallel to the x-axis, we can follow these steps:

### Step 1: Understand the condition for parallelism
A line that is parallel to the x-axis has a slope of 0. Therefore, we need to find the points on the curve where the derivative (slope) is equal to 0.

### Step 2: Differentiate the curve
We start by differentiating the function \( y = x + \frac{4}{x^2} \).

\[ \frac{dy}{dx} = \frac{d}{dx}(x) + \frac{d}{dx}\left(\frac{4}{x^2}\right) \]

Calculating the derivative of \( \frac{4}{x^2} \) using the power rule:

\[ \frac{d}{dx}\left(\frac{4}{x^2}\right) = 4 \cdot \frac{d}{dx}(x^{-2}) = 4 \cdot (-2)x^{-3} = -\frac{8}{x^3} \]

Thus, the derivative is:

\[ \frac{dy}{dx} = 1 - \frac{8}{x^3} \]

### Step 3: Set the derivative equal to 0
To find where the slope is 0, we set the derivative equal to 0:

\[ 1 - \frac{8}{x^3} = 0 \]

### Step 4: Solve for \( x \)
Rearranging the equation gives:

\[ \frac{8}{x^3} = 1 \]

Multiplying both sides by \( x^3 \):

\[ 8 = x^3 \]

Taking the cube root of both sides:

\[ x = 2 \]

### Step 5: Find the corresponding \( y \) value
Now that we have \( x = 2 \), we can find the corresponding \( y \) value on the curve:

\[ y = 2 + \frac{4}{2^2} = 2 + \frac{4}{4} = 2 + 1 = 3 \]

### Step 6: Conclusion
The value of \( k \) for the tangent line \( y = k \) that is parallel to the x-axis is:

\[ k = 3 \]

### Summary of Steps:
1. Identify that the slope of the tangent must be 0.
2. Differentiate the curve.
3. Set the derivative equal to 0 to find critical points.
4. Solve for \( x \).
5. Substitute \( x \) back into the original equation to find \( y \).
6. Conclude the value of \( k \).