If f(x)=(x3+b x2+c x+d)|(x-1)(x-2)(x-3)2(x-4)3| is derivable for all x ∈ R andf prime(3)+ f prime prime prime(4)=0 then | b + c + d | equals

Question

If f(x)=(x3+b x2+c x+d)|(x-1)(x-2)(x-3)2(x-4)3| is derivable for all x ∈ R andf prime(3)+ f prime prime prime(4)=0 then | b + c + d | equals (A) 0 (B) 1 (C) 2 (D) 3

Tardigrade Admin · Accepted Answer

∵ f prime prime prime(4)=0 ⇒( x -4) must be a factor of ( x 3+ bx 2+ cx + d ) and two other factors are (x-1) and (x-2) ∴ x 3+ bx 2+ cx + d =( x -1)( x -2)( x -4) 1+ b + c + d =0 ⇒| b + c + d |=1

Q. If $f (x) = (x^{3} + b x^{2} + c x + d) ∣ ∣ (x - 1) (x - 2) (x - 3)^{2} (x - 4)^{3} ∣ ∣$ is derivable for all $x \in R$ and $f^{'} (3) + f^{'''} (4) = 0$ then $∣ b + c + d ∣$ equals

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$∵ f^{'''} (4) = 0 \Rightarrow (x - 4)$ must be a factor of $(x^{3} + b x^{2} + c x + d)$ and two other factors are $(x - 1)$ and $(x - 2)$
$∴ x^{3} + b x^{2} + c x + d = (x - 1) (x - 2) (x - 4)$
$1 + b + c + d = 0 \Rightarrow ∣ b + c + d ∣ = 1$

Q. If f(x)=(x3+bx2+cx+d)∣∣​(x−1)(x−2)(x−3)2(x−4)3∣∣​ is derivable for all x∈R andf′(3)+f′′′(4)=0 then ∣b+c+d∣ equals

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1

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3

∵f′′′(4)=0⇒(x−4) must be a factor of (x3+bx2+cx+d) and two other factors are (x−1) and (x−2) ∴x3+bx2+cx+d=(x−1)(x−2)(x−4) 1+b+c+d=0⇒∣b+c+d∣=1

Q. If $f (x) = (x^{3} + b x^{2} + c x + d) ∣ ∣ (x - 1) (x - 2) (x - 3)^{2} (x - 4)^{3} ∣ ∣$ is derivable for all $x \in R$ and $f^{'} (3) + f^{'''} (4) = 0$ then $∣ b + c + d ∣$ equals

$∵ f^{'''} (4) = 0 \Rightarrow (x - 4)$ must be a factor of $(x^{3} + b x^{2} + c x + d)$ and two other factors are $(x - 1)$ and $(x - 2)$
$∴ x^{3} + b x^{2} + c x + d = (x - 1) (x - 2) (x - 4)$
$1 + b + c + d = 0 \Rightarrow ∣ b + c + d ∣ = 1$