Question

Assertion : Let $f : R \rightarrow R$ be a function such that $f ( x )= x ^{3}+ x ^{2}+3 x +\sin x$. Then $f$ is one-one.
Reason : $f ( x )$ neither increasing nor decreasing function.

• A. Assertion is correct, reason is correct; reason is a correct explanation for assertion.
• B. Assertion is correct, reason is correct; reason is not a correct explanation for assertion
• C. Assertion is correct, reason is incorrect
• D. Assertion is incorrect, reason is correct.

Answer 1

Answer: (C)

Every increasing or decreasing function is one-one
$f'( x )=3 x ^{2}+2 x +3+\cos\, x $
$=3\left( x +\frac{1}{3}\right)^{2}+\frac{8}{3}+\cos\, x > 0 $
${\left[\because|\cos\, x | < 1 \,\text{and} \, 3\left( x +\frac{1}{3}\right)^{2}+\frac{8}{3} \geq \frac{8}{2}\right]}$
$\therefore f ( x )$ is strictly increasing