Question

The minimum value of $2^{{\left(x^2-3\right)}^3}+27$ is

• A. $1$
• B. $2$
• C. $2^{27}$
• D. None of these

Answer 1

Answer: (A)

$2^{{\left(x^2-3\right)}^{3+27}}$ is minimum when ${\left(x^2-3\right)}^3+27$ is minimum Let $y={\left(x^2-3\right)}^3+27$
$=x^6-27-9x^4+27x^2+27$
$\Rightarrow y=x^6-9x^4+27x^2$
$\Rightarrow y=x^2\left(x^4-9x^2+27\right)$
$\Rightarrow y=x^2\left[x^4-9x^2+\frac{81}{4}-\frac{81}{4}+27\right]$
$\Rightarrow y=x^2\left[\left(x^4-9x^2+\frac{81}{4}\right)+\frac{27}{4}\right]$
$\Rightarrow y=x^2\left[{\left(x^2-\frac{9}{2}\right)}^2+\frac{27}{2}\right]\ge 0\forall x$
$\therefore$ Minimum value of ${\left(x^2-3\right)}^3+27$ is 0.
Hence, minimum value of $2^{{\left(x^2-3\right)}^3+27}$
$=2^0=1$