# How to find the minimum value of a function?

Now time to know **“How to find the minimum value of a function?”** The value of a function at a certain point is the output of the function when the input is equal to the point.

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## How to find the minimum value of a function?

There is no general answer to this question as it depends on the function in question. However, some methods for finding the minimum value of a function include using calculus to find the function’s derivative and setting it equal to zero or using a graphing calculator or computer software to find the function’s turning point.

Example

**Find the minimum value of**

f(x) = x^4 – 2x^2 + 1

We can use calculus to find the function’s derivative:

f ‘(x) = 4x^3 – 4x

Setting the derivative equal to zero and solving for x gives us the x-coordinates of the turning points:

0 = 4x^3 – 4x

4x(x^2 – 1) = 0

x = 0, ±1

Since we are only interested in the minimum value, we only need to consider the turning point at x = 1. We can use a graphing calculator or computer software to find that the y-coordinate of this turning point is f(1) = -1/4. Therefore, the minimum value of the function is -1/4.

We could also have just graphed the function to find its minimum value:

f(x) = x^4 – 2x^2 + 1

The graph of this function is shown below. The minimum value is approximately -0.5 and occurs at x = 1.

Read also: How to find the equation of a line with one point?

### What is the Value of a function formula?

The value of a function is the result of substituting a given number for the variable in the function formula.

For example, if f(x) = x2 + 3 and we want to know the value of f(2), we would substitute 2 for x in the function formula to get f(2) = 4 + 3 = 7.

Similarly, if we want to know the value of f(5), we would substitute 5 for x in the function formula to get f(5) = 25 + 3 = 28. And also

if we want to know the value of f(10), we would substitute 10 for x in the function formula to get f(10) = 100 + 3 = 103.

## How do you find the minimum or maximum value of a function?

To find the minimum or maximum value of a function, you must first find the critical points of the function. The critical points are the points where the derivative of the function is equal to zero or is undefined. Once you have found the critical points, you must then test the points around the critical points to see which points give the minimum or maximum value of the function.

To test the points around the critical points, you can use the second derivative test. If the second derivative of the function at a critical point is positive, then the critical point is a local minimum. If the second derivative of the function at a critical point is negative, then the critical point is a local maximum.

### 3 Ways to Find the Maximum or Minimum Value of a function

- Find the critical points of the function. These are the points where the derivative of the function is equal to zero or does not exist.
- At each critical point, determine whether the function is increasing or decreasing.
- The maximum value of the function will occur at the critical point where the function is decreasing. The minimum value of the function will occur at the critical point where the function is increasing.