Question

Assertion (A): Let $f:R\to R$ be a function such that $f\left(X\right)=X^3+X^2+3X+\sin X$, then $f$ is one to one .
Reason (R): $f\left(x\right)$ is neither increasing nor decreasing function.

• A. Both (A) and (R) are true and (R) is the correct explanation of (A).
• B. Both (A) and (R) are true and (R) is not the correct explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

Answer: (C)

(A) is true but (R) is false.

$f\left(X\right)=X^3+X^2+3X+\sin X$

$f^{\prime }\left(X\right)=3X^2+2X+3+\cos X$
$f^{\prime }\left(X\right)=3(X^2+\frac{2}{3}X+1)+\cos X$
$f^{\prime }\left(X\right)=3{(X+\frac{1}{3})}^2+\frac{8}{3}+\cos X$
Clearly, $f^{\prime }\left(X\right)>0$
Hence the function is monotonically increasing function and is one to one.
The assertion (A) is true but the reason (R) is false.