The function $f : [0, 3] \to [1, 29]$, defined by $f(x) = 2x^{3} - 15x^{2} + 36x + 1$, is

The function f : [0, 3] → [1, 29], defined by f(x) = 2x³ - 15x² + 36x + 1, is $f : [0, 3] \to [1, 29]$ $f(x) = 2x^3 - 15x^2 + 36x + 1$ $\text{To find: whether } f \text{ is one-one and/or onto}$ $f'(x) = 6x^2 - 30x + 36 = 6(x-2)(x-3)$ $x \in [0,2): f' > 0 \Rightarrow f \text{ is increasing}$ $x \in (2,3): f' < 0 \Rightarrow f \text{ is decreasing}$ $f(0) = 1, \quad f(2) = 16 - 60 + 72 + 1 = 29$ $f(3) = 54 - 135 + 108 + 1 = 28$ $\text{Range} = [1, 29] = \text{codomain} \Rightarrow \text{onto}$ $f(1.5) = 28 = f(3), \quad 1.5 \neq 3$ $\text{Same output for two inputs} \Rightarrow \text{not one-one}$ $\boxed{\text{onto but not one-one}} \quad \text{option (2)}$