Question

If the function $f\left(x\right)=x^4+b{x}^2+8x+1$ has a horizontal tangent and a point of inflection for the same value of $x,$ then the value of $b$ is equal to

• A. $-1$
• B. $1$
• C. $6$
• D. $-6$

Answer 1

The correct option is D $-6$

$f^{\prime }\left(x\right)=0$ and $f^{\prime \prime }\left(x\right)=0$ for the same $x=x_1$ (say)
Now, $f^{\prime }\left(x\right)=4x^3+2bx+8$
$f^{\prime }\left(x_1\right)=2(2x_1^3+b{x}_1+4)=0\dots \left(1\right)$
$f^{\prime \prime }\left(x_1\right)=2(6x_1^2+b)=0$
$\Rightarrow b=-6x_1^2$
Substituting this value of $b$ in $\left(1\right),$
$2x_1^3+(-6x_1^3)+4=0$
$\Rightarrow 4x_1^3=4$
$\Rightarrow x_1=1$
Hence, $b=-6$