The function f:0,7→0,70 where fx=x^3-12x^2+45x, is

Correct answer: (B) many-one & onto. f^(')x=3x^2-24x+45 =3x^2-8x+15=3x-3x-5 ∴f^(')x changes its sign for x∈0,7 ∴fx is many-one Now f0=0, f5=50, f3=54, f7=70 ∴Range∈0,70=co-domain ∴fx→onto

Q50 / 87 · NTA Abhyas (JEE Main) 2020 · Mathematics · Functions

The function $f : [0,7] \to [0, 70]$ where $f (x) = x^{3} - 12 x^{2} + 45 x$ , is

A
one-one & onto
Correct answer
B
many-one & onto
C
one-onto & into
D
many-one & into

Solution

f^{'} (x) = 3 x^{2} - 24 x + 45

= 3 (x^{2} - 8 x + 15) = 3 (x - 3) (x - 5)

∴ f^{'} (x)

changes its sign for

x \in [0,7]

∴ f (x)

is many-one
Now

f (0) = 0, f (5) = 50,

f (3) = 54, f (7) = 70

∴ R a n g e \in [0,70] = c o - d o m a i n

∴ f (x) \to o n t o

Hence, the correct answer is option: 'B'.

The function f:0,7→0,70 where fx=x3-12x2+45x, is

Solution

More Functions Questions

The function $f : [0,7] \to [0, 70]$ where $f (x) = x^{3} - 12 x^{2} + 45 x$ , is