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,So it is many one

Now at $x=0f\left(0\right)=1$ $x=2f\left(2\right)=16-60+72+1=29$
$x=3f\left(3\right)=54-135+108+1=28$

$f$ Has range $=[1,29]$
Hence given function is onto