# Maximum and Minimum value of a quadratic function

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
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 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 the 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 ` `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. `

Output:

```Maxvalue = 0.25
Minvalue = -Infinity
```

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