optimization find the point on the graph of the function that is closest to the given point f(X)= square root of x point:(8,0)
To find the point on the graph of the function that is closest to the given point (8, 0), we can use the concept of optimization.
Step 1: Distance Formula
Find the distance between any point (x, √(x)) on the graph of the function f(x) = √(x) and the given point (8, 0). We can use the distance formula:
Distance = √((x - 8)^2 + (√(x) - 0)^2)
Step 2: Minimize the Distance
Minimize the distance formula by finding the value of x that minimizes the distance function.
Step 3: Differentiate the Distance Function
Differentiate the distance function obtained from the distance formula with respect to x.
Step 4: Set the Derivative Equal to Zero
Set the derivative equal to zero and solve for x to find the critical points.
Step 5: Evaluate Critical Points
Evaluate the critical points to determine which point corresponds to the minimum distance.
Let's proceed with finding the point on the graph of the function that is closest to the given point (8, 0).
To find the point on the graph of the function that is closest to the given point, we can use the concept of optimization. Here's how you can approach this problem:
1. Start with the given function f(x) = √x and the given point (8,0).
2. To find the point on the graph of f(x) closest to the given point, we need to minimize the distance between the two points. The distance formula between two points (x1, y1) and (x2, y2) is given by:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
In our case, (x1, y1) = (8, 0), and (x2, y2) will be any point on the graph of f(x).
3. To minimize the distance, we can minimize the square of the distance, which is equivalent and simplifies the calculation. Thus, we can minimize the function:
g(x) = (x - 8)^2 + (f(x) - 0)^2
where f(x) = √x. This function represents the squared distance between any point (x, √x) on the graph of f(x) and the given point (8, 0).
4. Simplify g(x):
g(x) = (x - 8)^2 + (√x - 0)^2
= (x - 8)^2 + x
5. To find the minimum of g(x), we can differentiate g(x) with respect to x, set the derivative equal to zero, and solve for x:
g'(x) = 2(x - 8) + 1
= 2x - 15
2x - 15 = 0
2x = 15
x = 15/2 = 7.5
6. Now we have the x-coordinate of the point on the graph of f(x) that is closest to the given point. To find the corresponding y-coordinate, plug the value of x into the original function f(x) = √x:
f(7.5) = √(7.5) ≈ 2.7386
7. Therefore, the point on the graph of f(x) that is closest to the given point (8,0) is approximately (7.5, 2.7386).