Maximum value of `(sqrt(-3+4x-x^2)+4)^2+(x-5)^2` `(where 1 le x le 3)` is

Answer

To find the maximum value of the expression \((\sqrt{-3 + 4x - x^2} + 4)^2 + (x - 5)^2\) for \(1 \leq x \leq 3\), we will follow these steps:

### Step 1: Define the function
Let \(y = (\sqrt{-3 + 4x - x^2} + 4)^2 + (x - 5)^2\).

### Step 2: Determine the domain of the square root
The expression inside the square root, \(-3 + 4x - x^2\), must be non-negative:
\[ -3 + 4x - x^2 \geq 0. \]
Rearranging gives:
\[ x^2 - 4x + 3 \leq 0. \]
Factoring the quadratic:
\[ (x - 1)(x - 3) \leq 0. \]
This inequality holds for \(1 \leq x \leq 3\).

### Step 3: Find the critical points
To find the maximum value, we need to differentiate \(y\) with respect to \(x\) and set the derivative to zero.

First, we simplify \(y\):
\[ y = (\sqrt{-3 + 4x - x^2} + 4)^2 + (x - 5)^2. \]

Let \(u = \sqrt{-3 + 4x - x^2}\), then:
\[ y = (u + 4)^2 + (x - 5)^2. \]

### Step 4: Differentiate \(y\)
We will differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = 2(u + 4) \cdot \frac{du}{dx} + 2(x - 5). \]

To find \(\frac{du}{dx}\):
\[ u = \sqrt{-3 + 4x - x^2} \Rightarrow \frac{du}{dx} = \frac{1}{2\sqrt{-3 + 4x - x^2}}(4 - 2x). \]

Substituting \(\frac{du}{dx}\) back into \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = 2(\sqrt{-3 + 4x - x^2} + 4) \cdot \frac{1}{2\sqrt{-3 + 4x - x^2}}(4 - 2x) + 2(x - 5). \]

### Step 5: Set the derivative to zero
Setting \(\frac{dy}{dx} = 0\) will give us critical points. This may be complex, so we can also evaluate \(y\) at the endpoints \(x = 1\) and \(x = 3\).

### Step 6: Evaluate \(y\) at the endpoints
1. **At \(x = 1\)**:
\[ y(1) = (\sqrt{-3 + 4(1) - (1)^2} + 4)^2 + (1 - 5)^2 = (\sqrt{0} + 4)^2 + (-4)^2 = 4^2 + 16 = 16 + 16 = 32. \]

2. **At \(x = 3\)**:
\[ y(3) = (\sqrt{-3 + 4(3) - (3)^2} + 4)^2 + (3 - 5)^2 = (\sqrt{0} + 4)^2 + (-2)^2 = 4^2 + 4 = 16 + 4 = 20. \]

### Step 7: Compare values
Now we compare the values:
- \(y(1) = 32\)
- \(y(3) = 20\)

### Conclusion
The maximum value of the expression on the interval \(1 \leq x \leq 3\) is:
\[ \boxed{32}. \]