二次函数的最大值和最小值

给定二次函数ax ² + bx + c 。当x对所有可能的实数值都变化时，求出该函数可能的最大值和最小值。

例子：

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

方法：

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

该函数分为两部分。
第一部分是一个完美的平方函数。可能有两种情况：

1. 情况1：如果a的值为正。
   • 最大值等于无穷大。
   • 当第一部分等于零时，函数的最小值将出现，因为平方函数的最小值为零。
2. 情况2：如果a的值为负。
   • 最小值将等于-Infinity。
   • 由于为负，则任务以最大化负平方函数的负平方函数。再次最大值将等于零，因为这将是x的任何其他值的负值。

第二部分是给定二次函数的常数，因此对于任何x值都不能更改。因此，将在两种情况下都将其添加。因此，问题的答案是：

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

下面是上述方法的实现：

C++

// C++ implementation of the above approach
#include 
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.

输出：

Maxvalue = 0.25
Minvalue = -Infinity