Question

Assertion :$f\left(x\right)=a{x}^3+b{x}^2+cx+d\sin x$, then the condition that $f\left(x\right)$ is always one one function is $b^2<3a(c-\left|d\right|)$ Reason: For $f\left(x\right)$ to be one-one either $f$ is entirely increasing or entirely decreasing.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$f\left(x\right)=a{x}^3+b{x}^2+cx+d\sin x$
$\therefore$ $f^{\prime }\left(x\right)=3a{x}^2+2bx+c+d\cos x$
f(x) to be one-one function if $f^{\prime }\left(x\right)>0$
$\Rightarrow$ $3a{x}^2+2bx+c+d\cos x>0$
$\Rightarrow$ $3a{x}^2+2bx+c-\left|d\right|>0$
$\Rightarrow$ $4b^2-4\left(3a\right)(c-\left|d\right|)<0$ ($\because f\left(x\right)>0$, then $D<0$)
$\Rightarrow$ $b^2<3a(c-\left|d\right|)$