Question

The range of values of $k$ for which the function $f\left(x\right)=(k^2-4)x^2+6x^3+8x^4$ has a local maxima at point $x=0$ is

• A. $k<-2$ or $k>2$
• B. $k\le -2$ or $k\ge 2$
• C. $-2<k<2$
• D. $-2\le k\le 2$

Answer 1

The correct option is C $-2<k<2$

$f\left(x\right)=(k^2-4)x^2+6x^3+8x^4$
$f^{\prime }\left(x\right)=2x(k^2-4)+18x^2+32x^3$
$=2x[k^2-4+9x+16x^2]$
and $f^{\prime \prime }\left(x\right)=2k^2-8+36x+96x^2$
$\because x=0$ is a point of local maxima
$\Rightarrow f^{\prime \prime }\left(0\right)<0\Rightarrow 2k^2-8<0\Rightarrow (k-2)(k+2)<$
$0\Rightarrow -2<k<2$