## Minimum value

### In mathematics, finding the minimum value refers to finding the smallest possible value in a given set of numbers or a function

In mathematics, finding the minimum value refers to finding the smallest possible value in a given set of numbers or a function. The process of finding the minimum value depends on the context in which it is being used. Here, we will discuss two common scenarios: finding the minimum value in a set of numbers and finding the minimum value of a function.

1. Finding the Minimum Value in a Set of Numbers:

To find the minimum value in a set of numbers, you need to compare each number in the set and identify the smallest one. This is a straightforward process using the following steps:

Step 1: Identify the set of numbers for which you are finding the minimum value.

Step 2: Compare each number in the set with the others. Begin with the first number and compare it with the second number. If the first number is smaller, it becomes the current minimum. Otherwise, the second number becomes the current minimum.

Step 3: Repeat the comparison process for the remaining numbers in the set. If a number is smaller than the current minimum, update it as the new minimum.

Step 4: Continue this process until you have compared all the numbers in the set.

Step 5: The number that remains as the current minimum after all the comparisons is the minimum value in the set.

For example, let’s find the minimum value in the set {3, 5, 7, 2, 9}:

Step 1: The set of numbers is {3, 5, 7, 2, 9}.

Step 2: Compare the first number, 3, with the second number, 5. Since 3 is smaller, it becomes the current minimum.

Step 3: Compare the current minimum, 3, with the next number, 7. 3 is still the minimum.

Step 4: Compare the current minimum, 3, with the next number, 2. Since 2 is smaller, it becomes the new minimum.

Step 5: Compare the current minimum, 2, with the last number, 9. 2 is still the minimum.

Step 6: After comparing all the numbers, the minimum value is 2.

Therefore, the minimum value in the set {3, 5, 7, 2, 9} is 2.

2. Finding the Minimum Value of a Function:

To find the minimum value of a function, you need to identify the critical points of the function and determine which point corresponds to the minimum value. Here are the general steps:

Step 1: Identify the function for which you are finding the minimum value.

Step 2: Take the derivative of the function and set it equal to zero. Solve this equation to find the critical points.

Step 3: Determine the second derivative of the function.

Step 4: Substitute the critical points into the second derivative. If the second derivative is positive at a critical point, it corresponds to a minimum value. If the second derivative is negative, it corresponds to a maximum value.

Step 5: Compare the function value at each critical point. The critical point with the smallest function value is the minimum value.

Note: If the function is defined over a closed interval, you also need to evaluate the function at the endpoints of the interval and include them in the comparison step.

For example, let’s find the minimum value of the function f(x) = x^2 – 4x + 5:

Step 1: The function is f(x) = x^2 – 4x + 5.

Step 2: Take the derivative of the function:

f'(x) = 2x – 4.

Set the derivative equal to zero:

2x – 4 = 0

2x = 4

x = 2.

The critical point is x = 2.

Step 3: Take the second derivative of the function:

f”(x) = 2.

Step 4: Substitute the critical point into the second derivative:

f”(2) = 2.

Since the second derivative is positive, the critical point x = 2 corresponds to a minimum value.

Step 5: Compare the function value at the critical point and the endpoints of the interval (if applicable).

Evaluate the function at x = 2:

f(2) = (2)^2 – 4(2) + 5

= 4 – 8 + 5

= 1.

Therefore, the minimum value of the function f(x) = x^2 – 4x + 5 is 1.

Remember, the approach to finding the minimum value may vary based on the specific problem or context, but these general steps should help guide you through the process.

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