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Maximum and Minimum value of a quadratic function
  • Last Updated : 22 Apr, 2021

Given a quadratic function ax2 + bx + c. Find the maximum and minimum value of the function possible when x is varied for all real values possible.

Examples:

Input: a = 1, b = -4, c = 4
Output:
Maxvalue = Infinity
Minvalue = 0
Quadratic function given is x2 -4x + 4
At x = 2, value of the function is equal to zero.

Input: a = -1, b = 3, c = -2
Output:
Maxvalue = 0.25
Minvalue = -Infinity

Approach: 

 Q(x)=ax2 + bx + c.
     =a(x + b/(2a))2      +     c-b2/(4a).
       first part             second part

The function is broken into two parts. 
The first part is a perfect square function. There can be two cases: 

  1. Case 1: If value of a is positive. 
    • The maximum value would be equal to Infinity.
    • The minimum value of the function will come when the first part is equal to zero because the minimum value of a square function is zero.
  2. Case 2: If value of a is negative. 
    • The minimum value would be equal to -Infinity.
    • Since a is negative, the task to maximize the negative square function. Again maximum value of a negative square function would be equal to zero as it would be a negative value for any other value of x.

The second part is a constant value for a given quadratic function and hence cannot change for any value of x. Hence, it will be added in both cases. Hence, the answer to the problem is:



If a > 0,
    Maxvalue = Infinity
    Minvalue = c - b2 / (4a)
If a < 0,
    Maxvalue = c - b2 / (4a)
    Minvalue = -Infinity

Below is the implementation of the above approach: 

C++




// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to print the Maximum and Minimum
// values of the quadratic function
void PrintMaxMinValue(double a, double b, double c)
{
 
    // Calculate the value of second part
    double secondPart = c * 1.0 - (b * b / (4.0 * a));
 
    // Print the values
    if (a > 0) {
 
        // Open upward parabola function
        cout << "Maxvalue = "
             << "Infinity\n";
        cout << "Minvalue = " << secondPart;
    }
    else if (a < 0) {
 
        // Open downward parabola function
        cout << "Maxvalue = " << secondPart << "\n";
        cout << "Minvalue = "
             << "-Infinity";
    }
    else {
 
        // If a=0 then it is not a quadratic function
        cout << "Not a quadratic function\n";
    }
}
 
// Driver code
int main()
{
    double a = -1, b = 3, c = -2;
 
    PrintMaxMinValue(a, b, c);
 
    return 0;
}

Java




// Java implementation of the above approach
import java.util.*;
 
class GFG
{
 
    // Function to print the Maximum and Minimum
    // values of the quadratic function
    static void PrintMaxMinValue(double a, double b, double c)
    {
 
        // Calculate the value of second part
        double secondPart = c * 1.0 - (b * b / (4.0 * a));
 
        // Print the values
        if (a > 0)
        {
 
            // Open upward parabola function
            System.out.print("Maxvalue = "
                    + "Infinity\n");
            System.out.print("Minvalue = " + secondPart);
        }
        else if (a < 0)
        {
 
            // Open downward parabola function
            System.out.print("Maxvalue = " + secondPart + "\n");
            System.out.print("Minvalue = "
                    + "-Infinity");
        }
        else
        {
 
            // If a=0 then it is not a quadratic function
            System.out.print("Not a quadratic function\n");
        }
    }
 
    // Driver code
    public static void main(String[] args)
    {
        double a = -1, b = 3, c = -2;
 
        PrintMaxMinValue(a, b, c);
    }
}
 
// This code is contributed by Rajput-Ji

Python3




# Python3 implementation of the above approach
 
# Function to print the Maximum and Minimum
# values of the quadratic function
def PrintMaxMinValue(a, b, c) :
 
    # Calculate the value of second part
    secondPart = c * 1.0 - (b * b / (4.0 * a));
     
    # Print the values
    if (a > 0) :
         
        # Open upward parabola function
        print("Maxvalue =", "Infinity");
        print("Minvalue = ", secondPart);
         
    elif (a < 0) :
         
        # Open downward parabola function
        print("Maxvalue = ", secondPart);
        print("Minvalue =", "-Infinity");
         
    else :
         
        # If a=0 then it is not a quadratic function
        print("Not a quadratic function");
 
# Driver code
if __name__ == "__main__" :
    a = -1; b = 3; c = -2;
 
    PrintMaxMinValue(a, b, c);
 
# This code is contributed by AnkitRai01

C#




// C# implementation of the above approach
using System;
 
class GFG
{
     
    // Function to print the Maximum and Minimum
    // values of the quadratic function
    static void PrintMaxMinValue(double a, double b, double c)
    {
 
        // Calculate the value of second part
        double secondPart = c * 1.0 - (b * b / (4.0 * a));
 
        // Print the values
        if (a > 0)
        {
 
            // Open upward parabola function
            Console.Write("Maxvalue = "
                    + "Infinity\n");
            Console.Write("Minvalue = " + secondPart);
        }
        else if (a < 0)
        {
 
            // Open downward parabola function
            Console.Write("Maxvalue = " + secondPart + "\n");
            Console.Write("Minvalue = "
                    + "-Infinity");
        }
        else
        {
 
            // If a=0 then it is not a quadratic function
            Console.Write("Not a quadratic function\n");
        }
    }
 
    // Driver code
    static public void Main ()
    {
        double a = -1, b = 3, c = -2;
        PrintMaxMinValue(a, b, c);
    }
}
 
// This code is contributed by ajit.

Javascript




<script>
 
// Javascript implementation of the above approach
 
// Function to print the Maximum and Minimum
// values of the quadratic function
function PrintMaxMinValue(a, b, c)
{
     
    // Calculate the value of second part
    var secondPart = c * 1.0 -
                    (b * b / (4.0 * a));
 
    // Print the values
    if (a > 0)
    {
 
        // Open upward parabola function
        document.write("Maxvalue = " +
                       "Infinity" + "<br>");
        document.write("Minvalue = " +
                       secondPart);
    }
    else if (a < 0)
    {
 
        // Open downward parabola function
        document.write("Maxvalue = " +
                       secondPart + "<br>");
        document.write("Minvalue = " +
                       "-Infinity");
    }
    else
    {
         
        // If a=0 then it is not a quadratic function
        document.write("Not a quadratic function\n");
    }
}
 
// Driver Code
var a = -1, b = 3, c = -2;
 
PrintMaxMinValue(a, b, c);
 
// This code is contributed by Ankita saini
 
</script>
Output: 
Maxvalue = 0.25
Minvalue = -Infinity

 

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