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

The correct option is D

-6

$f^{\prime }\left(x\right)=0and{f}^{\prime \prime }\left(x\right)=0forthesamex=x_1\left(say\right)now{f}^{\prime }\left(x\right)=4x^3+2bx+8f^{\prime }\left(x_1\right)=2[2x_1^3+b{x}_1+4]=\mathrm{0.......}\left(1\right)f^{\prime \prime }(x_1=2[6x_1^2+b]=\mathrm{0........}(2)$
from equation (2), we get b = -6 $x_1^2$
Substituting the value of b in (1)
$2x_1^3+(-6x_1{)}^3+4=0\Rightarrow 4x_1^3=4\Rightarrow x_1=1Hence,b=-6$