What is the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 18)?
We can use the formula for the distance between a point $(x, y)$ and the origin $(0,0)$:
$$d = \sqrt{x^2 + y^2}$$
To minimize this distance, we can minimize $d^2 = x^2 + y^2$. We know that the point $(x,y)$ lies on the graph of $y = \frac{1}{2}(x^2 - 18)$, so we can substitute this into the expression for $d^2$:
$$d^2 = x^2 + y^2 = x^2 + \left(\frac{1}{2}(x^2 - 18)\right)^2$$
Simplifying this expression gives:
$$d^2 = x^2 + \frac{1}{4}(x^4 - 36x^2 + 324)$$
To find the minimum value of $d^2$, we can take the derivative with respect to $x$ and set it equal to 0:
$$\frac{d}{dx} (x^2 + \frac{1}{4}(x^4 - 36x^2 + 324)) = 2x + \frac{1}{4}(4x^3 - 72x) = 0$$
Simplifying this equation gives:
$$5x^3 - 36x = 0$$
Which factors as:
$$x(5x^2 - 36) = 0$$
The solutions are $x=0$ and $x=\pm\frac{6\sqrt{5}}{5}$. To determine which one gives the minimum distance, we can check the second derivative:
$$\frac{d^2}{dx^2} (x^2 + \frac{1}{4}(x^4 - 36x^2 + 324)) = 2 + 3x^2$$
At $x=0$, this is positive, so $x=0$ gives a minimum. The corresponding $y$ value is:
$$y = \frac{1}{2}(0^2 - 18) = -9$$
So the closest point on the graph to the origin is $\left(0, -9\right)$, and the distance between them is:
$$d = \sqrt{0^2 + (-9)^2} = \boxed{9}$$
To find the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 18), we need to find the point on the graph that is closest to the origin.
1. Recall that the distance between two points (x1, y1) and (x2, y2) is given by the formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
2. Since the origin is (0, 0), we need to find the point (x, y) on the graph of y = 1/2*(x^2 - 18) that minimizes the distance.
3. Substitute y = 1/2*(x^2 - 18) into the distance formula:
distance = sqrt(x^2 + (1/2*(x^2 - 18))^2)
4. Simplify the expression inside the square root:
distance = sqrt(x^2 + (1/4)*(x^2 - 18)^2)
5. To minimize the distance, we can minimize the expression inside the square root. Taking the derivative and setting it equal to zero will give us the critical points.
6. Differentiate the expression with respect to x:
distance' = (2x + (1/2)*(x^2 - 18)*2x)/2sqrt(x^2 + (1/4)*(x^2 - 18)^2)
7. Simplify the derivative:
distance' = (2x(x + (1/2)*(x^2 - 18)))/2sqrt(x^2 + (1/4)*(x^2 - 18)^2)
= x(x + (1/2)*(x^2 - 18))/sqrt(x^2 + (1/4)*(x^2 - 18)^2)
8. Set the derivative equal to zero and solve for x:
x(x + (1/2)*(x^2 - 18))/sqrt(x^2 + (1/4)*(x^2 - 18)^2) = 0
9. The critical points occur when x = 0 or x + (1/2)*(x^2 - 18) = 0.
For x = 0, we have a point on the y-axis. Let's solve x + (1/2)*(x^2 - 18) = 0 for the other critical point.
10. Simplify the equation:
x + (1/2)*(x^2 - 18) = 0
2x + x^2 - 18 = 0
x^2 + 2x - 18 = 0
11. Factor the equation:
(x + 6)(x - 3) = 0
12. Set each factor equal to zero and solve for x:
x + 6 = 0 --> x = -6
x - 3 = 0 --> x = 3
13. We have three critical points: x = 0, x = -6, and x = 3.
14. Plug these x-values back into the equation y = 1/2*(x^2 - 18) to find the corresponding y-values:
For x = 0: y = 1/2*(0^2 - 18) = -9
For x = -6: y = 1/2*(-6^2 - 18) = -27
For x = 3: y = 1/2*(3^2 - 18) = -10.5
15. The three points on the graph corresponding to the critical points are (0, -9), (-6, -27), and (3, -10.5).
16. Finally, calculate the distance between the origin (0, 0) and each of these points to find the smallest distance:
- Distance to (0, -9): sqrt((0 - 0)^2 + (-9 - 0)^2) = 9
- Distance to (-6, -27): sqrt((-6 - 0)^2 + (-27 - 0)^2) = 27
- Distance to (3, -10.5): sqrt((3 - 0)^2 + (-10.5 - 0)^2) ≈ 10.8
17. The smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 18) is 9, which occurs at the point (0, -9).
To find the smallest distance between the origin (0,0) and a point on the graph of y = 1/2*(x^2 - 18), we can use the concept of the distance formula.
The distance between two points (x1, y1) and (x2, y2) in a two-dimensional plane can be found using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Since we want to find the smallest distance between the origin (0,0) and a point on the given graph, we can substitute x1 = 0 and y1 = 0 into the distance formula.
Thus, the formula for finding the distance d simplifies to:
d = sqrt((x2 - 0)^2 + (y2 - 0)^2)
= sqrt(x2^2 + y2^2)
Now, let's substitute the equation of the graph y = 1/2*(x^2 - 18) into this formula:
d = sqrt(x2^2 + (1/2*(x2^2 - 18))^2)
= sqrt(x2^2 + (1/4)*(x2^4 - 36x2^2 + 324))
= sqrt(x2^2 + 1/4*x2^4 - 9/2*x2^2 + 81)
To find the smallest distance, we need to minimize this function. One way to do this is by taking the derivative of the function with respect to x and setting it equal to zero to find the critical points.
Let's differentiate the function and solve for x:
d' = 2x + (1/2)*(x^3 - 18x) - 9x
= 2x + 1/2*x^3 - 9x - 9x
= 1/2*x^3 - 18x
Setting this derivative equal to zero gives:
1/2*x^3 - 18x = 0
Now we can solve this equation to find the critical points.
Since x = 0 is one of the solutions, we have:
x = 0, x = 6√2, x = -6√2
Next, we substitute these critical point values into the distance function and find the corresponding distances:
d(0) = sqrt(0^2 + 0^2) = 0
d(6√2) = sqrt((6√2)^2 + 1/4*(6√2)^4 - 9/2*(6√2)^2 + 81)
= sqrt(72 + 216 - 216 + 81)
= sqrt(153)
d(-6√2) = sqrt((-6√2)^2 + 1/4*(-6√2)^4 - 9/2*(-6√2)^2 + 81)
= sqrt(72 + 216 - 216 + 81)
= sqrt(153)
Now, we compare these distances and find the smallest one. In this case, we can see that the smallest distance is 0, which occurs when x = 0. Hence, the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 18) is 0.