Question

For all $a,bϵR$ 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

Answer: (B)

exactly one extremum

$f\left(x\right)=3x^4-4x^3+6x^2+ax+b{f}^{{}^{\prime }}\left(x\right)=12x^3-12x^2+12x+a$
$f^{{}^{\prime }}\left(x\right)$ is a cubic polynomial and bijective for all real numbers
Therefor, only for one value of $x{f}^{{}^{\prime }}\left(x\right)=0$
Hence only one extremum exits.