Question

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

Answer 1

(i) $f\left(x\right)=3x^5+40x^3+240x$

Diff wrt x

$f^{\prime }\left(x\right)=15x^4+120x^2+240$

for the fucntion to be increasing $f^{\prime }\left(x\right)$ must be greater than zero

$15x^4+120x^2+240>0$

$3x^4+24x^2+48>0$

$x^4+8x^2+16>0$

Hence given function $f\left(x\right)>0$

so it is an increasing function

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

Diff wrt x

$f^{\prime }\left(x\right)=12x^2-36x+27$

for the function to be increasing

$f^{\prime }\left(x\right)$ must be greater than zero

$12x^2-36x+27>0$

$4x^2-12x+9>0$

Hence given function $f\left(x\right)>0$

So it is an increasing function.