# Maximize a value for a semicircle of given radius

We are given a **semi circle** with radius **R.** We can take any point on the circumference let it be **P.**Now, from that point **P** draw two lines to the two sides of diameter. Let the lines be **PQ** and **PS**.

The task is to find the maximum value of expression **PS ^{2} + PQ ** for a given

**R.**

**Examples :**

Input : R = 1 Output : 4.25 (4*1^2 + 0.25) = 4.25 Input : R = 2 Output : 16.25 (4 * 2^2 + 0.25)= 16.25

**Let F = PS^2 + PQ. We know QS = 2R (Diameter of the semicircle)**

**->** We also know that **triangle PQS** will always be **right angled triangle** irrespective of the position of point P on the circumference of circle

1.)QS^2 = PQ^2 + PS^2 (Pythagorean Theorem) 2.) Adding and Subtracting PQ on the RHS QS^2 = PQ^2 + PS^2 + PQ - PQ 3.) Since QS = 2R 4*R^2 + PQ - PQ^2 = PS^2 + PQ => 4*R^2 + PQ - PQ^2 = F 4.) Using the concept of maxima and minima differentiating F with respect to PQ and equating it to 0 to get the point of maxima for F i.e. 1 - 2 * PQ = 0 => PQ = 0.5 5.) Now F will be maximum at F = 4*R^2 + 0.25

## C++

`// C++ program to find ` `// the maximum value of F ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to find the ` `// maximum value of F ` `double` `maximumValueOfF (` `int` `R) ` `{ ` ` ` `// using the formula derived for ` ` ` `// getting the maximum value of F ` ` ` `return` `4 * R * R + 0.25; ` `} ` ` ` `// Drivers code ` `int` `main() ` `{ ` ` ` `int` `R = 3; ` ` ` `printf` `(` `"%.2f"` `, maximumValueOfF(R)); ` ` ` `return` `0; ` `} ` |

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## JAVA

`// JAVA program to find ` `// the maximum value of F ` `import` `java.io.*; ` ` ` `class` `GFG ` `{ ` ` ` ` ` `// Function to find the ` ` ` `// maximum value of F ` ` ` `static` `double` `maximumValueOfF (` `int` `R) ` ` ` `{ ` ` ` ` ` `// using the formula derived for ` ` ` `// getting the maximum value of F ` ` ` `return` `4` `* R * R + ` `0.25` `; ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main (String[] args) ` ` ` `{ ` ` ` `int` `R = ` `3` `; ` ` ` `System.out.println(maximumValueOfF(R)); ` ` ` `} ` ` ` `} ` ` ` `// This code is contributed ` `// by anuj_67. ` |

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## Python3

`# python program to find ` `# the maximum value of F ` ` ` `# Function to find the ` `# maximum value of F ` `def` `maximumValueOfF (R): ` ` ` ` ` `# using the formula derived for ` ` ` `# getting the maximum value of F ` ` ` `return` `4` `*` `R ` `*` `R ` `+` `0.25` ` ` ` ` `# Drivers code ` `R ` `=` `3` `print` `(maximumValueOfF(R)) ` ` ` `# This code is contributed by Sam007. ` |

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## C#

`// C# program to find the ` `// maximum value of F ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` ` ` `// Function to find the ` ` ` `// maximum value of F ` ` ` `static` `double` `maximumValueOfF (` `int` `R) ` ` ` `{ ` ` ` ` ` `// using the formula derived for ` ` ` `// getting the maximum value of F ` ` ` `return` `4 * R * R + 0.25; ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main () ` ` ` `{ ` ` ` `int` `R = 3; ` ` ` `Console.WriteLine(maximumValueOfF(R)); ` ` ` `} ` ` ` `} ` ` ` `// This code is contributed by Sam007. ` |

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## PHP

`<?php ` `// PHP program to find the ` `// maximum value of F ` ` ` `// Function to find the ` `// maximum value of F ` `function` `maximumValueOfF (` `$R` `) ` `{ ` ` ` ` ` `// using the formula derived ` ` ` `// for getting the maximum ` ` ` `// value of F ` ` ` `return` `4 * ` `$R` `* ` `$R` `+ 0.25; ` `} ` ` ` `// Drivers code ` `$R` `= 3; ` `echo` `maximumValueOfF(` `$R` `); ` ` ` `// This code is contributed ` `// by anuj_67. ` `?> ` |

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**Output :**

36.25

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