Question

Prove that the following functions are increasing on R.
(i) f$\left(x\right)=$3$x^5$ + 40$x^3$ + 240$x$
(ii) $f\left(x\right)=4x^3-18x^2+27x-27$

Answer 1

(i)

$$\begin{gathered} f\left(x\right)=3x^5+40x^3+240x \\ f\text{'}\left(x\right)=15x^4+120x^2+240 \\ =15\left(x^4+8x^2+16\right) \\ =15{\left(x^2+4\right)}^2>0,\forall x\in R\left[\because 15>0\mathrm{and}{\left(x^2+4\right)}^2>0\right] \end{gathered}$$

So, f(x) is increasing on R.

(ii) $f\left(x\right)=4x^3-18x^2+27x-27$

$$\begin{gathered} f\text{'}\left(x\right)=12x^2-36x+27 \\ \Rightarrow f\text{'}\left(x\right)=3\left(4x^2-12x+9\right) \\ \Rightarrow f\text{'}\left(x\right)=3{\left(2x-3\right)}^2>0,\forall x\in R \end{gathered}$$

So, f(x) is increasing on R.