Verify the monotony of f without the use of derivative

Question

$f:\Bbb R \rightarrow (2,+\infty)$

and it is known that $$f^3(x)-12f(x)=3x-15$$ I need to verify f's monotony but without the use of the derivative so

Let there be a second function $$H(x)=x^3-12x \qquad \forall x \epsilon (2,+\infty)$$ so we got this $$H (f (x))=3x-15$$$$\forall x_1,x_2 \epsilon \Bbb R : x_1 \lt x_2 \Rightarrow 3x_1-15 \lt 3x_2-15 \Rightarrow H (f (x_1)) \lt H (f (x_2))$$ so I need to verify the monotony of $H $ but I cant do it without derivative.

Answer 1

We have

$$H(z)-H(y)=(z^3-12z)-(y^3-12y)=(z^3-y^3)-12(z-y)$$

Now

$$z^3-y^3=(z-y)(z^2+yz+y^2)$$

so

$$H(z)-H(y)=(z-y)[(z^2+yz+y^2)-12]>0$$

if $z>y>2$