Question

The function $f:[0,3] \rightarrow[1,29]$, defined by $f(x)=2 x^3-15 x^2+36 x+1$, is $(3,-1), 70]$

(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

SOLUTION : $F:[0,3] \rightarrow[1,29]$

$f(x)=2 x^3-15 x^2+36 x+1 \\$

$f(x)=6 x^2-30 x+36=6\left(x^2-5 x+6\right)=6(x-2)(x-3)$

in given domain function has local maxima, it is many-one

Now at $x=0 \quad f(0)=1$

$x=2  f(2)=16-60+72+1=29 \\$

$x=3  f(3)=54-135+108+1=163-135=28$

Has range $=[1,29]$

Hence given function is onto 

Hence Option B is Correct.