Mathematics - Functions Question with Solution | TestHub

Given: $\mathrm{f}(\mathrm{x})={\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}+\mathrm{d}\sin \mathrm{x}$

For $\mathrm{f}(\mathrm{x})$ to be one-one, f should be strictly monotonic i.e. either f is strictly increasing or strictly decreasing.

Differentiating (1), we get $f^{\prime }(x)=3a{x}^2+2bx+c+d\cos x$

$$\ge 3{\mathrm{ax}}^2+2\mathrm{bx}+\mathrm{c}-|\mathrm{d}|;\text{ for }|\cos \mathrm{x}|\le 1$$

If $\mathrm{f}(\mathrm{x})>0$ for all $\mathrm{x}\in \mathrm{R}$, we must have $\mathrm{a}>0$ and $4{\mathrm{b}}^2-4.3\mathrm{a}(\mathrm{c}-|\mathrm{d}|)<0$

(from the quadratic expression)

$\Rightarrow {\mathrm{b}}^2<3\mathrm{a}(\mathrm{c}-|\mathrm{d}|)$.

$\therefore \mathrm{f}(\mathrm{x})$ is strictly increasing if $\mathrm{a}>0$ and ${\mathrm{b}}^2<3\mathrm{a}(\mathrm{c}-|\mathrm{d}|)$.

Also. ${\mathrm{f}}^{\prime }(\mathrm{x})=3{\mathrm{ax}}^2+2\mathrm{bx}+\mathrm{c}+\mathrm{d}\cos \mathrm{x}$

$$\le 3{\mathrm{ax}}^2+2\mathrm{bx}+\mathrm{c}+|\mathrm{d}|$$

If ${\mathrm{f}}^{\prime }(\mathrm{x})<0$ for all $\mathrm{x}\in \mathrm{R},\mathrm{a}<0$ and $4{\mathrm{b}}^2-12\mathrm{a}(\mathrm{c}+|\mathrm{d}|)<0\Rightarrow {\mathrm{b}}^2<3\mathrm{a}(\mathrm{c}+|\mathrm{d}|)$. Hence the required condition is (i) $\mathrm{a}>0,{\mathrm{b}}^2<3\mathrm{a}(\mathrm{c}-|\mathrm{d}|)$ (ii) $\mathrm{a}<0,{\mathrm{b}}^2<3\mathrm{a}(\mathrm{c}-|\mathrm{d}|)$.