Question

For what values of $a,f\left(x\right)=-x^3+4a{x}^2+2x-5$ is decreasing $\forall x$.

• A. $(1,2)$
• B. $(3,4)$
• C. $R$
• D. No value of $a$

Answer 1

Answer: (D)

No value of $a$

$f\left(x\right)=-x^3+4a{x}^2+2x-5$
$f^{\prime }\left(x\right)=-3x^2+8ax+2$
So discriminant of $f^{\prime }\left(x\right)$ is $D=(8a{)}^2-4(-3\left)\right(2)=64a^2+24>0\forall a\in R$
Hence $f\left(x\right)$ is increasing for every real value of $a.$