Maximum Value
Let f(x) be a function with the domain D ⊂ R. Then f(x) is said to have attained the maximum value at a point ‘a’ if f(a) ≥ f(x) for all the values of x. In this case, ‘a’ is called the point of maximum and f(a) is known as the maximum value or the greatest value or the absolute maximum value of f(x).
Minimum Value
Let f(x) be a function with domain D ⊂ R. Then f(x) is said to attain the minimum value at a point ‘a’ if f(x) ≥ f(a). In such a case, the point ‘a’ is called the point of minimum and f(a) is known as the minimum value or the least value or the absolute minimum value of f(x).Let us first see the maximum and minimum values of some very basic functions
Example1
Find the maximum and the minimum values of y = f(x) = − x – 1 + 4.
y = f(x)  y = f(x)  y = f(x)  y = f(x)  y = f(x)  
= x  = x^{2}  =  = log_{e}x  = x−2  
Maximum  +∝  +∝  +∝  +∝  +∝ 
Minimum  −∝  0  −∝  −∝  0

Solution
We have f(x) = − x – 1 + 4 for all the real values of x.
x – 1 ≥ 0 for all real values of x
Hence, − x – 1 ≤ 0
x – 1 + 4 ≤ 4
So, the maximum value of y = 4
Now, f(x) = 4 ⇒ − x – 1 + 4 = 4 ⇒ x – 1 = 0
⇒ x = 1.
Hence, f(x) attains the maximum value 4 at x = 1.
To find out the minimum value, we can make − x−1 as small as possible by taking different values of x. So, the minimum value of y will be −∝.
x – 1 ≥ 0 for all real values of x
Hence, − x – 1 ≤ 0
x – 1 + 4 ≤ 4
So, the maximum value of y = 4
Now, f(x) = 4 ⇒ − x – 1 + 4 = 4 ⇒ x – 1 = 0
⇒ x = 1.
Hence, f(x) attains the maximum value 4 at x = 1.
To find out the minimum value, we can make − x−1 as small as possible by taking different values of x. So, the minimum value of y will be −∝.
Example2
What is the maximum and minimum value of y = f(x) = x + 3?
Solution
We have y = f(x) = x + 3
x + 3 ≥ 0
⇒ f(x) ≥ 0 for all x ε R
So the minimum value of f(x) is 0, which is attained at x = −3.
To find out the maximum value, it can be seen that y = f(x) = x + 3 can be made as big as possible. So, the maximum value of y is +∝.
x + 3 ≥ 0
⇒ f(x) ≥ 0 for all x ε R