Question

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

(i) f(x) = x³, x ∈ [-2, 2]

(ii) f (x) = sin x + cos x , x ∈ [0, π]

(iii) f(x) = 4x - \(\frac{1}{2}\)x², x [-2, \(\frac{9}{2}\)]

(iv) f (x) = (x – 1)² + 3, x ∈ [-3,1]

Answer 1

(i) f(x) = x³ ,x ∈ [- 2,2] 

f’(x) = 0 ⇒ 3x² = 0 

⇒ x = 0 for finding the absolute maximum and absolute minimum, 

we have to evaluate f (0), f (2), f (-2) 

F(0) = (0)³ = 0, F(2) = (2)³ = 8, 

F(-2) = (-2)³ = – 8 

Absolute maximum = 8 and 

Absolute minimum = -8 

∴ maximum at 2 is 8 and minimum at -2 is – 8.

(ii)

(iii)

(iv) f (x) = (x – 1)² + 3, x ∈ [-3,1] 

f'(x) = 2(x – 1) 

f’(x) = 0 ⇒ (x – 1) = 0 

⇒ x = 1 we will evaluate f (-3) and f(1) 

f(-3) = (-3 – 1)² + 3= 16 + 3 = 19 

f(1) = (0)² + 3 = 3 

Absolute maximum at x = -3 is 19 and 

Absolute minimum x = 1 is 3.