Question

Find the absolute maximum and minimum values of f, if any, on the given interval, and state where those values occur.
(a) f(1) = x^2-x-2;(-infinity,+infinity)
(b) f(x) = x^4+4x;(-infinity,+infinity)​

Answer 1

Given :

(a) f(x) = x² - x - 2 ; (-∞ , ∞)

(b) f(x) = x⁴ + 4x ; (-∞ , ∞)​

To find :

Absolute maximum and absolute minimum values of f.

Solution :

(a) f(x) = x² - x - 2 ; (-∞ , ∞)

f'(x) = 0

f'(x) = 2x - 1

2x - 1 = 0

2x = 1

x = 1/2

f(x) = x² - x - 2

f(x) = (1/2)² - 1/2 - 2

f(x) = 1/4 -1/2 -2

f(x) = -9/4

x         f(x)  

1/2     -9/4

Absolute minimum = -9/4

No absolute maximum, since x = 1/2 would give an absolute minimum but not an absolute maximum.

(b) f(x) = x⁴ + 4x ; (-∞ , ∞)​

f'(x) = 0

f'(x) = 4x³ + 4

4x³ + 4 = 0

x³ = -1

x = -1

f(x) = x⁴ + 4x

f(x) = (-1)⁴ + 4(-1)

f(x) = 1 - 4

f(x) = -3

x         f(x)  

-1         -3

Absolute minimum = -3

No absolute maximum, since x = -1 would give an absolute minimum but not an absolute maximum.