Thank you for reporting, we will resolve it shortly
Q.
For all $a , b \in R$ the function $f ( x )=3 x ^4-4 x ^3+6 x ^2+ ax + b$ has -
Application of Derivatives
Solution:
$f(x)=3 x^4-4 x^3+6 x^2+a x+b$
$f^{\prime}(x)=g(x)=12 x^3-12 x^2+12 x+a$
$f^{\prime \prime}(x)=36 x^2-24 x+12$
$=12\left(3 x^2-2 x+1\right)$
$f ^{\prime \prime}( x ) >0$
$f ^{\prime \prime}( x )$ is increasing
$\Rightarrow f ^{\prime}( x )=0$ at exactly one point.
$\Rightarrow$ The given function has exactly one extremum.