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.
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
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:
- 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.
- Case 2: If value of a is negative.
- The minimum value would be equal to -Infinity.
- Since a is negative, the task to 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:
Maxvalue = 0.25 Minvalue = -Infinity
- Sum of first N terms of Quadratic Sequence 3 + 7 + 13 + ...
- Program to find the Roots of Quadratic equation
- Roots of the quadratic equation when a + b + c = 0 without using Shridharacharya formula
- Program to find number of solutions in Quadratic Equation
- Number of subarrays whose minimum and maximum are same
- Maximum and Minimum Values of an Algebraic Expression
- Find maximum and minimum distance between magnets
- Find the maximum possible value of the minimum value of modified array
- Minimum and maximum possible length of the third side of a triangle
- Break a number such that sum of maximum divisors of all parts is minimum
- Minimum and maximum number of N chocolates after distribution among K students
- Product of all Subsequences of size K except the minimum and maximum Elements
- Maximum and minimum sums from two numbers with digit replacements
- Minimum number of elements to be removed to make XOR maximum
- Minimum and Maximum element of an array which is divisible by a given number k
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.