For all a , b ∈ R the function f ( x )=3 x 4-4 x 3+6 x 2+ ax + b has -

Question

For all a , b ∈ R the function f ( x )=3 x 4-4 x 3+6 x 2+ ax + b has - (A) no extremum (B) exactly one extremum (C) exactly two extremum (D) three extremum

Tardigrade Admin · Accepted Answer

f(x)=3 x4-4 x3+6 x2+a x+b f prime(x)=g(x)=12 x3-12 x2+12 x+a f prime prime(x)=36 x2-24 x+12 =12(3 x2-2 x+1) f prime prime( x ) >0 f prime prime( x ) is increasing ⇒ f prime( x )=0 at exactly one point. ⇒ The given function has exactly one extremum.

Q. For all a,b∈R the function f(x)=3x4−4x3+6x2+ax+b has -

no extremum

exactly one extremum

exactly two extremum

three extremum

f(x)=3x4−4x3+6x2+ax+b f′(x)=g(x)=12x3−12x2+12x+a f′′(x)=36x2−24x+12 =12(3x2−2x+1) f′′(x)>0 f′′(x) is increasing ⇒f′(x)=0 at exactly one point. ⇒ The given function has exactly one extremum.

Q. For all $a, b \in R$ the function $f (x) = 3 x^{4} - 4 x^{3} + 6 x^{2} + a x + b$ has -