If (u,v) are the coordinates of the point on the curve `x^(3) =y (x-4)^(2)` where the ordinate is minimum then uv is equal to

Answer

To solve the problem, we need to find the coordinates (u, v) of the point on the curve \( x^3 = y (x - 4)^2 \) where the ordinate (y-coordinate) is minimum. We will then calculate the product \( uv \).

### Step-by-Step Solution:

1. **Express y in terms of x**:
The given equation is:
\[ y = \frac{x^3}{(x - 4)^2} \]

2. **Differentiate y with respect to x**:
We will use the quotient rule for differentiation. If \( y = \frac{f(x)}{g(x)} \), then:
\[ \frac{dy}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \]
Here, \( f(x) = x^3 \) and \( g(x) = (x - 4)^2 \).

- Calculate \( f'(x) \) and \( g'(x) \):
\[ f'(x) = 3x^2 \]
\[ g'(x) = 2(x - 4) \]

- Now applying the quotient rule:
\[ \frac{dy}{dx} = \frac{3x^2 (x - 4)^2 - x^3 \cdot 2(x - 4)}{(x - 4)^4} \]

3. **Set the derivative equal to zero**:
To find the critical points, we set the numerator equal to zero:
\[ 3x^2 (x - 4)^2 - 2x^3 (x - 4) = 0 \]
Factor out common terms:
\[ x^2 (x - 4) \left( 3(x - 4) - 2x \right) = 0 \]
This gives us:
\[ x^2 = 0 \quad \text{or} \quad x - 4 = 0 \quad \text{or} \quad 3(x - 4) - 2x = 0 \]

Solving these:
- \( x^2 = 0 \) gives \( x = 0 \)
- \( x - 4 = 0 \) gives \( x = 4 \)
- \( 3(x - 4) - 2x = 0 \) simplifies to \( 3x - 12 - 2x = 0 \) or \( x = 12 \)

4. **Determine which critical point gives a minimum**:
We have critical points at \( x = 0, 4, 12 \). We will use the first derivative test to determine the nature of these points.

- Test intervals around the critical points:
- For \( x < 0 \), \( \frac{dy}{dx} > 0 \)
- For \( 0 < x < 4 \), \( \frac{dy}{dx} < 0 \)
- For \( 4 < x < 12 \), \( \frac{dy}{dx} > 0 \)
- For \( x > 12 \), \( \frac{dy}{dx} > 0 \)

This indicates that \( x = 4 \) is a local maximum and \( x = 12 \) is a local minimum.

5. **Calculate the y-coordinate at x = 12**:
Substitute \( x = 12 \) into the equation for y:
\[ y = \frac{12^3}{(12 - 4)^2} = \frac{1728}{8} = 216 \]

6. **Identify u and v**:
Thus, the coordinates are \( u = 12 \) and \( v = 216 \).

7. **Calculate \( uv \)**:
\[ uv = 12 \times 216 = 2592 \]

### Final Answer:
The value of \( uv \) is \( 2592 \).