The maximum value of
`y = sqrt((x-3)^(2)+(x^(2)-2)^(2))-sqrt(x^(2)-(x^(2)-1)^(2))` is

Options

A. 3
B. `sqrt(10)`
C. `2sqrt(5)`
D. none of these

Explanation

To find the maximum value of the function

\[ y = \sqrt{(x-3)^2 + (x^2 - 2)^2} - \sqrt{x^2 - (x^2 - 1)^2} \]

we will analyze the two components of the equation step by step.

### Step 1: Simplify the second term

The second term is

\[ \sqrt{x^2 - (x^2 - 1)^2}. \]

Let's simplify it:

\[ (x^2 - 1)^2 = x^4 - 2x^2 + 1. \]

Thus,

\[ x^2 - (x^2 - 1)^2 = x^2 - (x^4 - 2x^2 + 1) = x^2 - x^4 + 2x^2 - 1 = -x^4 + 3x^2 - 1. \]

So, we rewrite the second term:

\[ \sqrt{-x^4 + 3x^2 - 1}. \]

### Step 2: Rewrite the entire expression

Now, we can rewrite the function \(y\):

\[ y = \sqrt{(x-3)^2 + (x^2 - 2)^2} - \sqrt{-x^4 + 3x^2 - 1}. \]

### Step 3: Analyze the first term

The first term, \(\sqrt{(x-3)^2 + (x^2 - 2)^2}\), represents the distance between the points \(A(x, x^2)\) and \(B(3, 2)\).

### Step 4: Identify points

Let’s identify the points:
- Point A: \(A(x, x^2)\)
- Point B: \(B(3, 2)\)
- Point C: \(C(0, 1)\)

### Step 5: Use distance formula

The distance \(AB\) is given by:

\[ AB = \sqrt{(x - 3)^2 + (x^2 - 2)^2}. \]

The distance \(AC\) is given by:

\[ AC = \sqrt{(x - 0)^2 + (x^2 - 1)^2}. \]

### Step 6: Find maximum value

To find the maximum value of \(y\), we can analyze the expression \(AB - AC\). The maximum value occurs when the distance \(BC\) is maximized.

### Step 7: Calculate distance BC

The distance \(BC\) is given by:

\[ BC = \sqrt{(3 - 0)^2 + (2 - 1)^2} = \sqrt{9 + 1} = \sqrt{10}. \]

### Conclusion

Thus, the maximum value of \(y\) is:

\[ \sqrt{10}. \]

### Answer

The maximum value of \(y\) is \(\sqrt{10}\).

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