Question

Let $f:R\to R$ defined by
$f(x)=2x^3+2x^2+300x+5sinx$, then $f$ is

• A. one-one onto
• B. one-one into
• C. many one onto
• D. many one into

Answer 1

Answer: (A)

$f(x)=2x^3+2x^2+300x+5sinx$
$f^{\prime }(x)=6x^2+4x+300+5cosx$
$=2(3x^2+2x+147)+(6+5cosx)$
Since $-1\le cosx\le 1\forall x\in R$
$\therefore 6+5cosx>0$
let $g(x)=3x^2+2x+147$
Since $a=3>0$ and $D=(2{)}^2-4\times 3\times 147<0$
$\therefore g(x)>0$
$f^{\prime }(x)>0$
$\therefore f(x)$ is an increasing function therefore said to be
$1-1$ function.
Now, $f(\infty )=-\infty$ and $f(\infty )=\infty$
$\therefore f(x)$ is continuous also.
$\therefore$ Range of $f=$ codomain of $f=R$.
$\therefore f$ is onto.