Question

The smallest positive integral value of $a$ , such that the function $f\left(x\right)=x^4-4a{x}^2+10$ has more than two local extrema, is

• A. $1$
• B. $2$
• C. $4$
• D. $16$

Answer 1

Answer: (A)

$f^{{}^{\prime }}\left(x\right)=4x^3-8ax$
$=4x\left(x^2-2a\right)$
$=4x\left(x-\sqrt{2a}\right)\left(x+\sqrt{2a}\right)$
Hence, $f\left(x\right)$ has $3$ local extrema if $\sqrt{2a}$ is real i.e., $2a>0\Rightarrow a>0$
$\therefore$ Value of minimum $a=1$