How to Find the Minimum Value of a Function | Calculus Methods and Examples

minimum value

In mathematics, the minimum value refers to the smallest or lowest value that a function or set of numbers can take

In mathematics, the minimum value refers to the smallest or lowest value that a function or set of numbers can take. It is the point or value where the function reaches its lowest point or the set of numbers has its smallest element.

To find the minimum value of a function, we can employ different methods depending on the context. In calculus, one common approach is to find the derivative of the function and identify the critical points where the derivative is equal to zero or undefined. After computing these critical points, we evaluate the function at these points and compare their values to determine which one is the minimum value.

For example, let’s say we have a function f(x) = x^2 – 2x + 1. To find the minimum value, we take the derivative of f(x) with respect to x:

f'(x) = 2x – 2.

Next, we set the derivative equal to zero and solve for x:

2x – 2 = 0
2x = 2
x = 1.

Now we evaluate the function at x = 1:

f(1) = (1)^2 – 2(1) + 1
= 1 – 2 + 1
= 0.

Therefore, the minimum value of the function f(x) = x^2 – 2x + 1 occurs at x = 1 and has a value of 0.

It is important to note that sometimes a function may not have a minimum value if it is unbounded below (e.g., y = x in the set of real numbers). In other cases, the minimum value may be approached but not actually attained.

In summary, the minimum value is the smallest value of a function or set of numbers and can be found by analyzing critical points, evaluating the function at those points, and comparing the values.

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