Okay, so I need to find the smallest distance between the origin and a point on the graph of y = (1/√2)(x² - 3). The answer should be expressed as √a / b, where a and b are positive integers, and a isn't divisible by the square of any integer greater than one. Then, I have to find a + b. Hmm, let's start by recalling how to find the distance from a point to a graph.

First, the distance from the origin (0,0) to a point (x, y) on the graph is given by the distance formula: √(x² + y²). Since y is defined by the equation y = (1/√2)(x² - 3), I can substitute that into the distance formula. So the distance D becomes √(x² + [(1/√2)(x² - 3)]²). 

To find the smallest distance, I need to minimize D. However, minimizing the square of the distance, D², is equivalent because the square root function is monotonically increasing. So, instead of dealing with the square root, I can focus on minimizing D² = x² + [(1/√2)(x² - 3)]². Let's compute that expression.

First, expand the squared term: [(1/√2)(x² - 3)]² = (1/2)(x² - 3)². So, D² becomes x² + (1/2)(x² - 3)². Let's write that out: D² = x² + (1/2)(x⁴ - 6x² + 9). Now, let's expand that: D² = x² + (1/2)x⁴ - 3x² + (9/2). Combine like terms: (1/2)x⁴ + (x² - 3x²) + 9/2 = (1/2)x⁴ - 2x² + 9/2.

So, D² = (1/2)x⁴ - 2x² + 9/2. To find the minimum of this function, I can take the derivative with respect to x, set it equal to zero, and solve for x. Let's compute the derivative:

d(D²)/dx = 2*(1/2)x³ - 4x = x³ - 4x.

Set the derivative equal to zero: x³ - 4x = 0. Factor out x: x(x² - 4) = 0. So, the critical points are x = 0, x = 2, and x = -2. Now, we need to check which of these gives the minimum distance.

Let's evaluate D² at each critical point:

1. x = 0:
D² = (1/2)(0)⁴ - 2(0)² + 9/2 = 0 - 0 + 9/2 = 9/2. So, D = √(9/2) = 3/√2 = (3√2)/2. But let's note this value.

2. x = 2:
Compute D² = (1/2)(16) - 2(4) + 9/2 = 8 - 8 + 4.5 = 4.5 = 9/2. So, D = 3/√2 again.

3. x = -2:
Same as x = 2 because the function is even in x. So, D² is also 9/2. So, all three critical points give the same D². Wait, that's interesting. So, the minimal distance squared is 9/2, so the minimal distance is 3/√2. But 3/√2 can be rationalized as (3√2)/2. So, in the form √a / b, that would be √(9*2)/2 = √18 / 2. But √18 simplifies to 3√2, so that's the same as before. But the question says a should not be divisible by the square of any integer greater than one. So, we need to express √a / b such that a is square-free. 18 factors into 2*3², so the square part is 3². Therefore, to make a square-free, we need to factor out the square. So, √18 = √(9*2) = 3√2. Therefore, √18 / 2 = 3√2 / 2. So, 3√2 / 2 is already simplified with a square-free a. Wait, but the problem says "the smallest distance can be expressed as √a / b". So, if we have 3√2 / 2, that's equivalent to √(9*2) / 2 = √18 / 2. But 18 is divisible by 9, which is a square. So, to make a square-free, we need to write it as √(2*3²)/2 = 3√2 / 2. But that's not in the form √a / b. Hmm, maybe I made a mistake here.

Wait, maybe I need to rationalize the denominator? Let's see. 3/√2 is equal to (3√2)/2. So, if I write it as √(9*2)/2, that's √18 / 2, but 18 is not square-free. Alternatively, if I write it as 3√2 / 2, then the numerator is 3√2, which is not a single square root. So, perhaps I need to consider the form √a / b where a is square-free. Therefore, 3√2 / 2 can be written as √(9*2)/2 = √18 / 2, but since 18 is not square-free, maybe I need to adjust. Wait, perhaps there's a different approach.

Alternatively, maybe I messed up the calculation. Let's check the derivative again. The derivative of D² is x³ - 4x. Setting to zero gives x(x² - 4) = 0, so x = 0 or x = ±2. Then, substituting back into y:

For x = 0: y = (1/√2)(0 - 3) = -3/√2. So, the point is (0, -3/√2). The distance from origin is √(0 + (9/2)) = √(9/2) = 3/√2.

For x = 2: y = (1/√2)(4 - 3) = 1/√2. The point is (2, 1/√2). Distance squared is 4 + (1/2) = 4.5 = 9/2, same as before. Similarly for x = -2, same distance.

So, all three points give the same minimal distance. So, the minimal distance is 3/√2. To express this as √a / b, we have 3/√2 = √(9)/√2 = √(9/2). Wait, but 9/2 is 4.5, which is not an integer. The problem states that a and b are positive integers, so perhaps we need to rationalize the denominator.

So, 3/√2 = (3√2)/2. Therefore, this is equal to √(9*2)/2 = √18 / 2. But 18 is not square-free. So, to make a square-free, we have to write it as 3√2 / 2. But this is not in the form √a / b where a is square-free. Hmm, so maybe the problem expects a to be square-free, so 3√2 / 2 is acceptable because a is 2, but then 3 is outside the square root. Wait, but the problem says "the smallest distance can be expressed as √a / b". So, if we have 3√2 / 2, that's equivalent to √(9*2)/2, which is √18 / 2. But 18 is 9*2, so 9 is a square. So, to make a square-free, we need to have a = 2 and b = 2, but then 3 is outside. Hmm, maybe this is a point of confusion.

Wait, perhaps the problem allows a to be a multiple of a square? But the problem states that a is not divisible by the square of any integer greater than one. So, a must be square-free. Therefore, we need to express the distance as √a / b where a is square-free. Therefore, 3√2 / 2 can be expressed as √(9*2)/2, but since 9 is a square, that's not allowed. So, perhaps we need to manipulate it differently.

Wait, 3√2 / 2 is already simplified with a square-free a. Because 3 is outside the square root, and the square root is of 2, which is square-free. So, perhaps the problem allows the expression to have a coefficient multiplied by a square root, but the a inside the square root must be square-free. Let me check the problem statement again: "can be expressed as √a / b, where a and b are positive integers such that a is not divisible by the square of any integer greater than one." So, the expression is √a over b, with a square-free. So, 3√2 / 2 is equal to √(9*2)/2, but that would require a = 18, which is not square-free. Alternatively, 3√2 / 2 is equal to (√2 * 3)/2, which is √2 multiplied by 3/2. But the problem says the expression is √a / b. So, perhaps we need to write it as √( (3√2)^2 ) / 2 = √18 / 2, but again, 18 is not square-free. 

Wait, maybe I made a mistake in the initial substitution. Let me verify the distance calculation again. The distance squared is x² + y², with y = (1/√2)(x² - 3). So, y² = (1/2)(x² - 3)^2. Then, D² = x² + (1/2)(x^4 - 6x² + 9) = x² + (1/2)x^4 - 3x² + 9/2 = (1/2)x^4 - 2x² + 9/2. Correct.

Taking derivative: d(D²)/dx = 2*(1/2)x³ - 4x = x³ - 4x. Correct. Setting to zero: x(x² - 4) = 0. So, x=0, x=2, x=-2. Correct.

At x=0: y = -3/√2. Distance squared: 0 + (9/2) = 9/2. So, distance is 3/√2. Correct.

At x=2: y = (1/√2)(4 - 3) = 1/√2. Distance squared: 4 + (1/2) = 4.5 = 9/2. So, same as before.

So, all critical points give the same distance. Therefore, the minimal distance is indeed 3/√2, which is equal to √(9*2)/2 = √18 / 2. But since 18 is not square-free, we need to rationalize the denominator.

Alternatively, maybe the problem allows the expression to have a coefficient in front of the square root, like 3√2 / 2, which is √(9*2)/2, but written as 3√2 / 2. Since a is supposed to be square-free, 2 is square-free, so maybe this is acceptable. Let me check the problem statement again: "can be expressed as √a / b, where a and b are positive integers such that a is not divisible by the square of any integer greater than one." So, the numerator is √a, not multiplied by anything. Therefore, 3√2 / 2 is not in the form √a / b. Instead, it's 3 times √2 divided by 2, which is different. Therefore, to write it as √a / b, we need to have a single square root term. So, we need to manipulate it so that the numerator is just a square root.

So, 3√2 / 2 = √(9*2) / 2 = √18 / 2. But 18 is not square-free. Therefore, this seems like a problem. Wait, maybe I made a mistake in the problem statement. Let me check again: "the smallest distance between the origin and a point on the graph of y = (1/√2)(x² - 3) can be expressed as √a / b, where a and b are positive integers such that a is not divisible by the square of any integer greater than one."

Wait, so if the expression is √a / b, then 3√2 / 2 is not of that form. So, we need to express 3√2 / 2 as √a / b, which would require that 3√2 / 2 = √a / b. Then, squaring both sides: (9*2)/4 = a / b². So, 18/4 = a / b² => 9/2 = a / b². Therefore, a = (9/2) b². Since a and b are integers, 2 must divide b². Therefore, b must be even. Let b = 2k, then a = (9/2)(4k²) = 18k². But a must not be divisible by the square of any integer greater than one. So, a = 18k². But 18k² is divisible by 9 (which is 3²), so unless k = 1 and a = 18, which is divisible by 9. That's not allowed. Therefore, there's a contradiction here.

Wait, that suggests that it's impossible to express 3√2 / 2 as √a / b with a square-free. But that can't be, because the problem says it can be. Therefore, perhaps I made a mistake in the distance calculation.

Wait, let's recheck. The distance squared is D² = x² + y² = x² + [ (1/√2)(x² - 3) ]². Let's compute that again. [ (1/√2)(x² - 3) ]² = (1/2)(x² - 3)^2. So, D² = x² + (1/2)(x^4 - 6x² + 9). Then, expanding: (1/2)x^4 - 3x² + 9/2 + x² = (1/2)x^4 - 2x² + 9/2. So, correct.

Taking derivative: d(D²)/dx = 2*(1/2)x³ - 4x = x³ - 4x. Setting to zero: x(x² - 4) = 0. Correct. So, x = 0, ±2. Then, substituting back, we get the same distance. Therefore, the minimal distance is indeed 3/√2. But 3/√2 is equal to 3√2 / 2, which is not in the form √a / b unless a = 18, which is not square-free. Therefore, maybe I need to rationalize it differently.

Wait, perhaps the problem allows a to be multiplied by a square? But no, the problem states that the expression is √a / b. So, maybe the answer is 3√2 / 2, which is equivalent to √(9*2)/2 = √18 / 2. But since 18 is not square-free, we need to write it as √(9*2)/2 = 3√2 / 2. Therefore, maybe the problem allows a to be 2 and b to be 2/3, but b must be an integer. So, that approach doesn't work. Alternatively, perhaps I made a mistake in the initial substitution.

Wait, maybe there's another approach. Let's consider using calculus of variations or parametric equations, but since it's a function of x, the previous method is correct. Alternatively, maybe using geometry, but the distance formula is the standard approach here.

Alternatively, maybe the minimal distance occurs where the line from the origin is perpendicular to the tangent of the curve. That might be another way, but since we've already found the critical points via calculus, which gives the minimal distance as 3/√2, perhaps we need to accept that and express it in the required form.

But according to the problem statement, the answer is supposed to be √a / b with a square-free. So, 3/√2 is equal to 3√2 / 2, which is √(9*2)/2. But 9*2 is 18, which is not square-free. Therefore, perhaps there's a different minimal distance. Wait, but all critical points give the same distance. So, maybe the minimal distance is indeed 3/√2, which is 3√2 / 2, but since 18 isn't square-free, maybe the problem has a typo or I'm misunderstanding something.

Wait, let me check the problem statement again: "the smallest distance between the origin and a point on the graph of y=1/√2(x² - 3) can be expressed as √a / b, where a and b are positive integers such that a is not divisible by the square of any integer greater than one. Find a + b."

So, maybe I made a mistake in the distance formula. Let me verify again. The distance squared is x² + y², where y = (1/√2)(x² - 3). So, y² = (x² - 3)^2 / 2. Then, D² = x² + (x² - 3)^2 / 2. Let's compute this:

x² + [(x² - 3)^2]/2

Let me expand (x² - 3)^2: x^4 - 6x² + 9. Then, divided by 2: (x^4 - 6x² + 9)/2. So, D² = x² + (x^4 - 6x² + 9)/2 = (2x² + x^4 - 6x² + 9)/2 = (x^4 - 4x² + 9)/2. So, D² = (x^4 - 4x² + 9)/2. Then, derivative is d(D²)/dx = (4x³ - 8x)/2 = 2x³ - 4x. Wait, hold on! This contradicts the previous derivative.

Wait, earlier I took the derivative of D² = (1/2)x^4 - 2x² + 9/2. Then, derivative is x³ - 4x. But now, if D² = (x^4 - 4x² + 9)/2, then derivative is (4x³ - 8x)/2 = 2x³ - 4x. Which is different from before. Wait, so which one is correct?

Wait, let's recast. Original D² = x² + [(1/√2)(x² - 3)]². Which is x² + (1/2)(x² - 3)^2. Then, expanding (x² - 3)^2: x^4 - 6x² + 9. Then, multiply by 1/2: (x^4 - 6x² + 9)/2. Then, add x²: x² + (x^4 - 6x² + 9)/2 = (2x² + x^4 - 6x² + 9)/2 = (x^4 - 4x² + 9)/2. So, D² = (x^4 - 4x² + 9)/2. Therefore, derivative is (4x³ - 8x)/2 = 2x³ - 4x. So, setting to zero: 2x³ - 4x = 0 => 2x(x² - 2) = 0. Therefore, x = 0 or x = ±√2. Wait, so earlier I had x³ - 4x = 0, leading to x = 0, ±2. But now, with the correct expansion, derivative is 2x³ - 4x = 0, leading to x = 0, ±√2. So, there's a discrepancy here. Which one is correct?

Wait, this is confusing. Let's check step by step. The distance squared is D² = x² + y², where y = (1/√2)(x² - 3). Therefore, y = (x² - 3)/√2. Therefore, y² = (x² - 3)^2 / 2. So, D² = x² + [(x² - 3)^2]/2. Then, expanding (x² - 3)^2: x^4 - 6x² + 9. Then, divided by 2: (x^4 - 6x² + 9)/2. Then, add x²: x² + (x^4 - 6x² + 9)/2 = (2x² + x^4 - 6x² + 9)/2 = (x^4 - 4x² + 9)/2. Therefore, D² = (x^4 - 4x² + 9)/2. Then, derivative is (4x³ - 8x)/2 = 2x³ - 4x. Therefore, setting to zero: 2x³ - 4x = 0 => 2x(x² - 2) = 0 => x = 0 or x² = 2 => x = 0 or x = ±√2. Therefore, the critical points are x = 0, x = ±√2. Wait, so earlier when I computed the derivative, I must have made a mistake. Let's check again.

Originally, I thought that D² = (1/2)x^4 - 2x² + 9/2. Then, derivative is x³ - 4x. But according to this step-by-step, D² is (x^4 - 4x² + 9)/2. Therefore, derivative is 2x³ - 4x. So, there was a miscalculation earlier. Therefore, the critical points are x = 0, ±√2. Let's substitute these back into the distance formula.

For x = 0: y = (1/√2)(0 - 3) = -3/√2. The point is (0, -3/√2). Distance squared is 0 + (9/2) = 9/2. So, distance is 3/√2.

For x = √2: y = (1/√2)( (√2)^2 - 3 ) = (1/√2)(2 - 3) = (1/√2)(-1) = -1/√2. The point is (√2, -1/√2). Distance squared is (√2)^2 + (-1/√2)^2 = 2 + (1/2) = 2.5 = 5/2. So, distance is √(5/2) = √(10)/2 ≈ 1.58. Wait, that's smaller than 3/√2 ≈ 2.12. Therefore, this is the minimal distance?

Wait, but according to the critical points, x = ±√2 gives the minimal distance? Wait, but when we computed the derivative earlier, we had two different results. Let's check which one is correct.

Wait, the problem is that when expanding (x² - 3)^2, which is x^4 - 6x² + 9. Then, dividing by 2 gives (x^4 - 6x² + 9)/2. Then, adding x² gives x² + (x^4 - 6x² + 9)/2 = (2x² + x^4 - 6x² + 9)/2 = (x^4 - 4x² + 9)/2. Therefore, derivative is (4x³ - 8x)/2 = 2x³ - 4x. Therefore, critical points at x = 0 and x = ±√2.

But when we first computed D² as (1/2)x^4 - 2x² + 9/2, which is equivalent to (x^4 - 4x² + 9)/2, so that's correct. Therefore, the critical points are x = 0, ±√2.

But when we plug x = 0, we get distance 3/√2 ≈ 2.1213. When we plug x = ±√2, we get distance √(5/2) ≈ 1.5811. Therefore, the minimal distance is √(5/2), which is approximately 1.58. Therefore, the minimal distance squared is 5/2, so D = √(5/2) = √10 / 2. Therefore, expressed as √a / b, a = 10, b = 2. Since 10 is square-free (its prime factors are 2 and 5, neither of which is squared), so a = 10, b = 2. Therefore, a + b = 12.

Wait, so earlier I had a mistake in the derivative calculation. The critical points are at x = ±√2, not at x = ±2. Therefore, the minimal distance is √10 / 2, so a = 10, b = 2, and a + b = 12. So, the answer is 12.

But why did the initial derivative calculation differ? Because when I first computed D², I mistakenly thought that expanding (x² - 3)^2 / 2 and then adding x² would give (1/2)x^4 - 2x² + 9/2, which is incorrect. Actually, expanding (x² - 3)^2 gives x^4 - 6x² + 9. Then, divided by 2 gives (x^4 - 6x² + 9)/2. Adding x² gives (x^4 - 4x² + 9)/2. Therefore, derivative is 2x³ - 4x. Setting to zero gives x = 0, ±√2. Therefore, the minimal distance is at x = ±√2, giving D = √(5/2) = √10 / 2. So, a = 10, b = 2, a + b = 12.

Therefore, the answer is 12.

But let me verify once more. Let's compute D² at x = √2:

D² = (√2)^2 + y² = 2 + [ (1/√2)( (√2)^2 - 3 ) ]^2 = 2 + [ (1/√2)(2 - 3) ]^2 = 2 + [ (-1/√2) ]^2 = 2 + (1/2) = 2.5 = 5/2. Therefore, D = √(5/2) = √10 / 2. Yes, that's correct. So, the minimal distance is √10 / 2. Therefore, expressed as √a / b, a = 10, b = 2. Since 10 is square-free (prime factors 2 and 5, no squares), so a + b = 12.

Therefore, the answer is 12.

But why did the previous critical point at x = 0 give a larger distance? Because x = 0 corresponds to the point (0, -3/√2), which is vertically below the origin, while the minimal distance occurs at x = ±√2, where the curve is less extreme. So, the minimal distance is indeed √10 / 2, which gives a + b = 12.

Therefore, the answer is 12.
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