To find the x-coordinate of the point on the parabola $f(x) = x^2 - 1$ that is closest to the origin, we will use the distance formula.

Step 1: Distance Formula

The distance $d$ between any point $(x, f(x))$ on the parabola and the origin $(0, 0)$ is given by the distance formula:

$d = \sqrt{x^2 + (f(x))^2}$

Here, $f(x) = x^2 - 1$, so:

$d = \sqrt{x^2 + (x^2 - 1)^2}$

Step 2: Minimize the Distance

To minimize the distance, we can minimize the square of the distance (which avoids dealing with the square root):

$d^2 = x^2 + (x^2 - 1)^2$

Expand $(x^2 - 1)^2$:

$d^2 = x^2 + (x^4 - 2x^2 + 1)$

Simplify:

$d^2 = x^4 - x^2 + 1$

Step 3: Take the Derivative

Now, to find the value of $x$ that minimizes $d^2$, we take the derivative of $d^2$ with respect to $x$:

$\frac{d}{dx}(d^2) = \frac{d}{dx}(x^4 - x^2 + 1) = 4x^3 - 2x$

Step 4: Set the Derivative Equal to Zero

Set the derivative equal to zero to find the critical points:

$4x^3 - 2x = 0$

Factor the equation:

$2x(2x^2 - 1) = 0$

This gives us two possible solutions:

$x = 0 \quad \text{or} \quad 2x^2 - 1 = 0$

Solve $2x^2 - 1 = 0$:

$2x^2 = 1 \quad \Rightarrow \quad x^2 = \frac{1}{2} \quad \Rightarrow \quad x = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}$

Step 5: Check Which is the Minimum

Now, we have three critical points: $x = 0$, $x = \frac{\sqrt{2}}{2}$, and $x = -\frac{\sqrt{2}}{2}$.

We need to determine which of these points is closest to the origin. Since the distance formula involves a squared term, the values for $d^2$ at each of these points will give us the actual distances (because distance is always positive):

1. At $x = 0$: $d^2 = 0^4 - 0^2 + 1 = 1$

2. At $x = \frac{\sqrt{2}}{2}$: $d^2 = \left( \frac{\sqrt{2}}{2} \right)^4 - \left( \frac{\sqrt{2}}{2} \right)^2 + 1 = \frac{2}{4} - \frac{2}{4} + 1 = 1$

3. At $x = -\frac{\sqrt{2}}{2}$: $d^2 = \left( -\frac{\sqrt{2}}{2} \right)^4 - \left( -\frac{\sqrt{2}}{2} \right)^2 + 1 = 1$

We see that the distances for all these points are equal to 1, but since $x = 0$ is the simplest and closest to the origin (it’s the vertex of the parabola), it is the closest point.

Final Answer:

The x-coordinate of the point on the parabola $f(x) = x^2 - 1$ that is closest to the origin is $x = 0$.

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Would you like further details on any part of this solution?

Here are 5 related questions you might find interesting:

1. How do you find the minimum of a function using derivatives?
2. What is the significance of the second derivative in optimization problems?
3. How can we use the distance formula in 3-dimensional space?
4. What is the geometric interpretation of the derivative of a function at a point?
5. How do we solve optimization problems involving constraints?

Tip: When minimizing or maximizing a function, always check for critical points (where the derivative is zero) and confirm they are minima or maxima using either the second derivative test or by analyzing the behavior of the function.