Question

For all a, b $\in R$ the function $f\left(x\right)=3x^4-4x^3+6x^2+ax+b$ has

• A. no extremum
• B. exactly one extremum
• C. exactly two extremum
• D. three extremum

Answer 1

The correct option is B exactly one extremum

$f\left(x\right)=3x^4-4x^3+6x^2+ax+b$
$f^{\prime }\left(x\right)=g\left(x\right)=12x^3-12x^2+12x+a$
$f"\left(x\right)=36x^2-24x+12=12(3x^2-2x+1)$
Clearly $f"\left(x\right)>0$
$f^{\prime }\left(x\right)$ is increasing and it is cubic polynomial therefore it have at most one real root.
$\Rightarrow f^{\prime }\left(x\right)=0$ at exactly one point.
$\Rightarrow$ The given function has exactly one extremum.