Question

If the function $f(x)=x^4+b{x}^2+8x+1$ 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

Answer: (D)

$f^{\prime }(x)=0\text{ and }{f}^{\prime \prime }(x)=0\text{ for the same }x=x_1\text{ (say) }$
$\text{ now }{f}^{\prime }(x)=4x^3+2bx+8$
$f^{\prime }\left(x_1\right)=2\left[2x_1^3+bx{x}_1+4\right]=0\dots .(1)$
$f^{\prime \prime }\left(x_1\right)=2\left[6x_1^2+b\right]=0$.....(2)
from (2) $b=-6x_1^2$
substituting this value of $b$ in (1)
$2x_1^3+\left(-6x_1^3\right)+4=0\Rightarrow 4x_1^3=4\Rightarrow x_1=1\text{. Hence }b=-6$