Question

For what real values of a and b are all the extrema of the function $f\left(x\right)=a^2{x}^3+a{x}^2-x+b$ negative and, the maximum is at the point $x_0=-1$?

Answer 1

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

$f^{\prime }\left(x\right)=3a^2{x}^2+2ax-1=0$

$f^{\prime }(-1)=3a^2-2a-1=0\Rightarrow (3a+1)(a-1)=0$

$a=1,-\frac{1}{3}$

$f\left(x\right)<0$

For a=1

$f(-1)=-1+1+1+b<0\Rightarrow b<(-1)\Rightarrow b\in (-\infty ,-1)$

For $a=-\frac{1}{3}$

$f(-1)=-\frac{1}{9}-\frac{1}{3}+1+b<0\Rightarrow b<-\frac{5}{9}\Rightarrow b\in (-\infty ,-\frac{5}{9})$