Let (a,b) be a point on the parabola `y=4x-x^(2)` and is the point nearest to the point A(-1,4) Find (a+b).

Answer

To solve the problem, we need to find the point \((a, b)\) on the parabola \(y = 4x - x^2\) that is nearest to the point \(A(-1, 4)\). We will then compute \(a + b\).

### Step-by-Step Solution:

1. **Identify the relationship between \(a\) and \(b\)**:
Since the point \((a, b)\) lies on the parabola, we can express \(b\) in terms of \(a\):
\[ b = 4a - a^2 \]

2. **Set up the distance formula**:
The distance \(D\) between the point \((a, b)\) and the point \(A(-1, 4)\) can be expressed as:
\[ D = \sqrt{(a + 1)^2 + (b - 4)^2} \]
To minimize the distance, we can minimize \(D^2\) (since the square root function is increasing):
\[ D^2 = (a + 1)^2 + (b - 4)^2 \]

3. **Substitute \(b\) in the distance formula**:
Substitute \(b = 4a - a^2\) into the distance squared:
\[ D^2 = (a + 1)^2 + (4a - a^2 - 4)^2 \]

4. **Expand the distance squared**:
First, expand \((a + 1)^2\):
\[ (a + 1)^2 = a^2 + 2a + 1 \]
Now expand \((4a - a^2 - 4)^2\):
\[ 4a - a^2 - 4 = -a^2 + 4a - 4 \]
\[ (-a^2 + 4a - 4)^2 = (a^2 - 4a + 4)^2 = (a^2 - 4a + 4)(a^2 - 4a + 4) \]
Expanding this gives:
\[ = a^4 - 8a^3 + 24a^2 - 32a + 16 \]

5. **Combine the terms**:
Now combine the terms in \(D^2\):
\[ D^2 = a^2 + 2a + 1 + a^4 - 8a^3 + 24a^2 - 32a + 16 \]
\[ = a^4 - 8a^3 + 25a^2 - 30a + 17 \]

6. **Differentiate and find critical points**:
To find the minimum distance, differentiate \(D^2\) with respect to \(a\):
\[ \frac{d(D^2)}{da} = 4a^3 - 24a^2 + 50a - 30 \]
Set the derivative equal to zero to find critical points:
\[ 4a^3 - 24a^2 + 50a - 30 = 0 \]
Dividing through by 2:
\[ 2a^3 - 12a^2 + 25a - 15 = 0 \]

7. **Use the Rational Root Theorem**:
Testing \(a = 1\):
\[ 2(1)^3 - 12(1)^2 + 25(1) - 15 = 2 - 12 + 25 - 15 = 0 \]
So \(a = 1\) is a root.

8. **Factor the cubic polynomial**:
Using synthetic division or polynomial long division, we can factor out \((a - 1)\):
\[ 2a^3 - 12a^2 + 25a - 15 = (a - 1)(2a^2 - 10a + 15) \]

9. **Solve the quadratic**:
Now solve \(2a^2 - 10a + 15 = 0\) using the quadratic formula:
\[ a = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 2 \cdot 15}}{2 \cdot 2} \]
\[ = \frac{10 \pm \sqrt{100 - 120}}{4} = \frac{10 \pm \sqrt{-20}}{4} \]
The discriminant is negative, indicating that the other roots are imaginary.

10. **Find \(b\)**:
Since \(a = 1\) is the only real solution, substitute \(a\) back to find \(b\):
\[ b = 4(1) - (1)^2 = 4 - 1 = 3 \]

11. **Calculate \(a + b\)**:
Finally, compute \(a + b\):
\[ a + b = 1 + 3 = 4 \]

### Final Answer:
\[ \boxed{4} \]