Question

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

Answer 1

(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.