Local Minimum of the Function f(x) = (x/2) * (2/x)

To find the local minimum of the function f(x) = (x/2) * (2/x), we need to determine the critical points and then analyze the behavior of the function around those points.

Finding the Critical Points:
A critical point occurs where the derivative of the function is equal to zero or undefined. Let's find the derivative of f(x) first:

f'(x) = [(1/2) * (2/x)] - [(x/2) * (2/x^2)]

Simplifying the derivative, we get:

f'(x) = (1/x) - (1/x^2)

To find the critical points, we set the derivative equal to zero and solve for x:

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

Multiplying through by x^2, we have:

x - 1 = 0

Solving for x, we find x = 1.

Analyzing the Behavior:
Now let's analyze the behavior of the function around the critical point x = 1 to determine if it is a local minimum.

1. Left Side of x = 1:
For values of x less than 1, the function f(x) is positive. As x approaches 1 from the left side, the function values increase without bound. Therefore, there is no local minimum on the left side of x = 1.

2. Right Side of x = 1:
Similarly, for values of x greater than 1, the function f(x) is negative. As x approaches 1 from the right side, the function values decrease without bound. Therefore, there is no local minimum on the right side of x = 1.

Conclusion:
Since there is no local minimum on either side of x = 1, we can conclude that x = 1 is indeed a local minimum of the function f(x) = (x/2) * (2/x).

Visualization:
To visualize the behavior of the function and confirm the local minimum at x = 1, we can plot the graph of f(x) = (x/2) * (2/x). The graph will show a curve with a downward concavity near x = 1, indicating a local minimum at that point.

(Note: You can use graphing tools or software to plot the graph and verify the local minimum at x = 1.)

References:
- EduRev. (n.d.). Retrieved from

https://www.edurev.in/ https://www.edurev.in/