Question

If $f\left(x\right)=x^3-6x^2+6x$ then $f(2^{1/3}+2^{2/3}+2)$ has the value

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (A)

$2$

Let $x=2^{1/3}+2^{2/3}+2$
$\Rightarrow (x-2)=2^{1/3}+2^{2/3}$
Cubing both sides,
$(x-2{)}^3=(2^{1/3}+2^{2/3}{)}^3$
$\Rightarrow x^3-8-6x(x-2)=2+2^2-3\cdot 2(2^{1/3}+2^{2/3}-2)$
$[\because (a+b{)}^3=a^3+b^3+3ab(a+b)]$
$\Rightarrow x^3-8-6x(x-2)=6-6(x-2)$
$\Rightarrow x^3-6x^2+6x=14-12=2$
Hence option 'A' is correct choice.