In mathematics, maxima and minima, known collectively as extrema (singular: extremum), are the largest value (maximum) or smallest value (minimum), that a function takes in a point either within a given neighbourhood (local extremum) or on the function domain in its entirety (global or absolute extremum). More generally, the maxima and minima of a set (as defined in set theory) are the greatest and least values in the set. Source – Wikipedia.
Steps to Find Maximum and Minimum value:
Find the maximum and minimum value of a function in calculus, the following rule has to be followed,
Step1: Find the f '(x), which is the delineation of the function f(x).
Step 2: Solve f '(x) =0, Find the roots a, b, c, etc. Then these are the applicants for maximum and minimum values.
Step 3: We examine the function at everyone of these values. Let us take x =c.
Step 4: Establish the symbol of f ' (x) for values of 'x' faintly less than 'c' and for 'x' faintly greater than 'c'.
Conclusion:
If f '(x) symbol modifies from positive to negative as 'x' raises. Then x =c is the point of maximum value.
If f '(x) symbol modifies from negative to negative as 'x' raises. Then x =c is the point of minimum value.
If f ' (x) sign not modifies as 'x' increases. Then x =c is the point of neither maximum nor minimum value.
Example problems for minimum and maximum value:
1) Find the maximum value of y = -3x^2 + x – 5
Solution:-
Y= -3x^2 + x – 5
= -3(x^2 – 1/3) x – 5
= -3(x – 1/6)^2 + 1/12– 5
11
= -3(x – 1/6)^2 – 4 12
11
Maximum value is – 4 12.
2) Find two nonnegative values which addition is 7 and so that the product of one value and the square of the other value is a maximum.
Solution:
Consider variables x and y represent two nonnegative numbers. The sum of the two numbers are given to be
7 = x + y ,
So that
y = 7 - x .
We desire to maximize the product
P = x y^2.
But, before we differentiate the right side, we will write it as a function of x only. Substitute for y getting
P = xy^2
= x (7-x)^2.
Now differentiate this equation using the product law and chain rule, getting
P' = x(2) (7-x)(-1) + (1) ( 7-x)^2
= (7-x) [-2x + (7-x)]
= (7-x) [7-3x]
= (7-x) (3)[ 3-x ]
= 0
for
x=7 or x=3 .
Note that because both x and y are nonnegative numbers and their sum is 9, it follows that 0 <=x<= 7. See the adjoining sign chart for P'.
If x = 3 and y= 6 ,
then
P= 100 is the largest product.
