Question

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

The correct option is B

Onto but not One-One

$F:[0,3]\to [1,29]$
$f\left(x\right)=2x^3-15x^2+36x+1$
$f^{\prime }\left(x\right)=6x^2-30x+36$
$=6(x^2-5x+6)$
$=6(x-2)(x-3)$
In given domain function has local maxima, it is many-one

now at,
$x=0$ $f\left(0\right)=1$
$x=2$ $f\left(2\right)=16-60+72+1=29$
$x=3$ $f\left(3\right)=54-135+108+1$
$=163-135=25$
has range $=[1,29]$
hence given function is onto.