Question

Find the values of a for which $f\left(x\right)=a{x}^3-9a{x}^2+9x+25$ is increasing on R.

Answer 1

Given $f\left(x\right)=a{x}^3-9a{x}^2+9x+25$

The given function is increasing on $R$

If $f^{\prime }\left(x\right)>0$ on $R$

$f^{\prime }\left(x\right)=3a{x}^2-18ax+9$

$=3(a{x}^2-6ax+3)>0$ for $x\in (-\infty ,\infty )$

The only possible case is that determinant is less than zero and coefficient of $x^2$ is positive.

$\Delta =9(36a^2-4.a.3)<0$

$\Rightarrow 3a^2-a<0$

$\Rightarrow a(3a-1)<0$

$\Rightarrow 0<a<\frac{1}{3}$.

and $3a>0\Rightarrow a>0$

$\therefore$ The value of a must be such that

$0<a<\frac{1}{3}$