Question

Let f(x) = x³ − 6x² + 15x + 3. Then,
(a) f(x) > 0 for all x ∈ R
(b) f(x) > f(x + 1) for all x ∈ R
(c) f(x) is invertible
(d) none of these

Answer 1

(c) f(x) is invertible
f(x) =x³ − 6x² + 15x + 3
$$\begin{gathered} f\text{'}\left(x\right)=3x^2-12x+15 \\ =3\left(x^2-4x+5\right) \\ =3\left(x^2-4x+4+1\right) \\ =3{\left(x-2\right)}^2+\frac{1}{3}>0 \\ \mathrm{Therefore},f\left(x\right)\mathrm{is}\mathrm{strictly}\mathrm{increasing}\mathrm{function}. \\ \Rightarrow f^{-1}\left(x\right)\mathrm{exists}. \\ \mathrm{Hence},f\left(x\right)\mathrm{is}\mathrm{an}\mathrm{invertible}\mathrm{function}. \end{gathered}$$