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) = (x)³ + 1

Solution:

Maxima and minima are known as the extrema of a function

(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

⇒ x = 1/2

Hence, minimum value of

f = f (1/2)

= [(2 (1/2)) - 1] + 3

= 3

Thus, the function f does not have a maximum value.

(ii) The given function is

f (x) = 9x² + 12x + 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

⇒ x = - 2/3

Therefore, Minimum value of

f = f (- 2/3)

= (3 (- 2/3) + 2)² - 2

= - 2

Hence, the 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

Therefore, Maximum value of

f = f (1)

= - (1 - 1)² + 10

= 10

Hence, the function f does not have a minimum value.

(iv) The given function is

g (x) = (x)³ + 1

Hence, function g neither has a maximum value nor a minimum value

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.5 Question 1

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) = (x)³ + 1

Summary:

(i)function f does not have a maximum value (ii)  function f does not have a maximum value. (iii)function f does not have a minimum value (iv) function g neither has a maximum value nor a minimum value