Minimum Values of a Function

Introduction

A function is a term used in mathematics for describing the relationship between any two or more given variables. This is usually used to define the independent and dependent variables’ relationship. In mathematical terms, if any given variable x is related to another variable y, then the numerical value of x is said to be a function of the independent variable y. Let’s get to know the minimum value of a function in detail.

Types of Function

Functions can be distinguished by various equations and algebraic expressions. They can be classified based on the degree of the polynomial.

• Constant Function – It is the polynomial function with the degree of zero
• Linear Function – It is the polynomial function with the degree one
• Quadratic Function – It is the polynomial function with the degree of two
• Cubic Function – It is the polynomial function with the degree of three

Minimum Value of a Function

The minimum value of any quadratic function is the point in the graphical representation that has a vertex at the lowest point. It is usually used in quadratic functions to find out the minimum cost or area and is practically used in science, architecture, and business.

Determination of Minimum Value

The minimum value of a quadratic equation can be determined by various methods.

By using Graph 

You can plot the values determined by the given equation in a graph and find out the minimum value by visually locating the minimum point on the graph. The minimum value is the y-value of the vertex of the graph.

Using a General Form of the Function

Let us study this method with a suitable example in a step-by-step format for easy understanding.

• Step 1 – Setting up the function

Any quadratic function has to contain the second-degree, which is the x2 term

Example

f(x)=3x+2x-x2+3x2+4

We know that the general format for a quadratic equation can be expressed as:

f(x)=ax2+bx+c

Now, let us combine the x2 terms and x terms

We get, f(x)=2x2+5x+4

• Step 2 – Determining the graph direction

Quadratic function graphs are usually represented in parabolic form.

The parabola can open upwards or downwards depending on the coefficient of the x2 term, which is denoted as a. If the value of a is found out to be positive, the parabola opens in an upward fashion.

If the value of a is found to be negative, the parabola opens in a downward fashion. The minimum value of the quadratic function can be found if the parabola opens upward. Likewise, the maximum value of the quadratic function can be found if the parabola opens downward.

Examples

If f(x)=2x2+4x-6, the value of a is 2, so the parabola opens upward.

If f(x)=-5x2+2x+8, the value of a is -5, so the parabola opens downward.

If f(x)=x2+6, the value of a is 1, so the parabola opens upward.

• Step 3 – Calculating -b/2a value

 The -b/2a value needs to be calculated to determine the x value of the vertex of the parabola.

Examples

1. If f(x)=x2+10x-1, then a = 1 and b = 10.

          x = -b/2a

          x = -10/2*1

          x = -10/2 = -5

This implies that the x value of the vertex is -5.

1. If f(x)=-3x2+6x-4, then a = -3 and b = 6.

            x = -b/2a

            x = -6/(2)(-3)

            x = -6/(-6) = 1

This implies that the x value of the vertex is 1.

• Step 4 – Finding the corresponding f(x) value

Once the value of x is known, it can be substituted in the equation to find the corresponding value of f(x). This gives us the minimum or maximum value of the function.

Examples

For f(x)=x2+10x-1, x = -5.

Substituting x value in the equation, we get,

f(-5)=(-5)2+10(-5)-1

f(-5)=25-50-1 = -26

For f(x)=-3x2+6x-4, x = 1

Substituting x value in the equation, we get,

f(1)=-3(1)2+6(1)-4

f(1)=-3+6-4 = -1

• Step 5 – Reporting the results

 Once we arrive at the solution, we can conclude by giving a result.

Examples

For f(x)=x2+10x-1, a value is positive, so the minimum value of the function has to be reported. Hence, the vertex is located at (-5,-26), and the minimum value is -26.

For f(x)=-3x2+6x-4, a value is negative, so the maximum value of the function has to be reported. Hence, the vertex is located at (1,-1), and the maximum value is -1.

Conclusion

Any function consisting of two variables can have a local maximum and a local minimum value. These maximum and minimum points of a function are collectively known as the extrema of the function. Critical points in the graph are present in the domain of the function in conditions where the derivative is equal to zero or are undefined. The various types of functions based on equations are constant function, linear function, quadratic function, and cubic function. These study material notes on minimum values of a function can help us better understand this concept and be useful for students preparing for competitive exams.