| Statement-1: Let \[f:R\to R\]be a function such that \[f(x)={{x}^{3}}+{{x}^{2}}+3x+\sin x.\]Then f is one-one. |
| Statement-2: f(x) neither increasing nor decreasing function. |
A) Statement-1 is false, Statement-2 is true
B) Statement-1 is true, Statement-2 is true, and Statement-2 is a correct explanation for Statement-1
C) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
D) Statement-1 is true, Statement-2 is false.
Correct Answer: D
Solution :
Every increasing or decreasing function is one-one \[f'(x)=3{{x}^{2}}+2x+3+\cos x=3{{\left( x+\frac{1}{3} \right)}^{2}}+\] \[\frac{8}{3}+\cos x>0\] \[[\because |\cos x|<|\]and\[3{{\left( x+\frac{1}{3} \right)}^{2}}+\frac{8}{3}\ge \frac{8}{3}]\] \[\therefore f(x)\] is strictly increasing
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