MCQ
The function $f:[0,3] \rightarrow[1,29]$, defined by $f(x)=2 x^3-15 x^2+36 x+1$, is
- Aone-one and onto
- ✓onto but not one-one
- Cone-one but not onto
- Dneither one-one nor onto
✓
Answer
Correct option: B.
onto but not one-one
b
$F:[0,3] \rightarrow[1,29] $
$F:[0,3] \rightarrow[1,29] $
$f(x)=2 x^3-15 x^2+36 x+1 $
$f^{\prime}(x)=6 x^2-30 x+36 $
$=6\left(x^2-5 x+6\right) $
$=6(x-2)(x-3)$
in given domain function has local maxima, it is many-one
Now at
$x=0 $$ f(0) $$ =1 $
$x=2 $$ f(2) $$ =16-60+72+1=29 $
$x $$ =3 $$ f(3) $$ =54-135+108+1 $
$ =163-135=28$
Has range $=[1,29]$
Hence given function is onto
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