Question

The function f : [0, 3] → [1, 29], defined by f(x) = 2x³ - 15x² + 36x + 1, is

(A)  one-one and onto

(B)  onto but not one-one 

(C)  one-one but not onto 

(D)  neither one-one nor onto

Answer 1

Correct option (B) onto but not one-one

Explanation :

f(x) = 2x³ - 15x² + 36x + 1

f'(x) = 6x² - 30x + 36

= − 6(x² - 5x + 6)

= 6(x - 2)(x - 3)

f(x) is increasing in [0, 2] and decreasing in [2, 3]. f (x) is many one.

f(0) = 1

f(2) = 29

f(3) = 28

Range is [1, 29].

Hence, f (x) is many-one-onto.