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Find the maximum and minimum values, if any, of the following functions given by
(i) f(x) = (2x − 1)2 + 3 (ii) f(x) = 9x2 + 12x + 2
(iii) f(x) = −(x − 1)2 + 10 (iv) g(x) = x3 + 1
Asked by Kalpana Last Modified
Answer 
Harshit Bhardwaj
(i) The given function is f(x) = (2x − 1)2 + 3.
It can be observed that (2x − 1)2 ≥ 0 for every x ∈ R.
Therefore, f(x) = (2x − 1)2 + 3 ≥ 3 for every x ∈ R.
The minimum value of f is attained when 2x − 1 = 0.
2x − 1 = 0 ⇒ 
∴Minimum value of f =
= 3
Hence, function f does not have a maximum value.
(ii) The given function is f(x) = 9x2 + 12x + 2 = (3x + 2)2 − 2.
It can be observed that (3x + 2)2 ≥ 0 for every x ∈ R.
Therefore, f(x) = (3x + 2)2 − 2 ≥ −2 for every x ∈ R.
The minimum value of f is attained when 3x + 2 = 0.
3x + 2 = 0 ⇒ 
∴Minimum value of f = 
Hence, function f does not have a maximum value.
(iii) The given function is f(x) = − (x − 1)2 + 10.
It can be observed that (x − 1)2 ≥ 0 for every x ∈ R.
Therefore, f(x) = − (x − 1)2 + 10 ≤ 10 for every x ∈ R.
The maximum value of f is attained when (x − 1) = 0.
(x − 1) = 0 ⇒ x = 1
∴Maximum value of f = f(1) = − (1 − 1)2 + 10 = 10
Hence, function f does not have a minimum value.
(iv) The given function is g(x) = x3 + 1.
Hence, function g neither has a maximum value nor a minimum value.
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