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Q.
If the function $f ( x )= x ^4+ bx ^2+8 x +1$ has a horizontal tangent and a point of inflection for the same value of $x$ then the value of $b$ is equal to
Application of Derivatives
Solution:
$ 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 )=4 x ^3+2 bx +8 $
$ f ^{\prime}\left( x _1\right)=2\left[2 x _1^3+ bx x _1+4\right]=0 \ldots .(1)$
$ f ^{\prime \prime}\left( x _1\right)=2\left[6 x _1^2+ b \right]=0$.....(2)
from (2) $b=-6 x_1^2$
substituting this value of $b$ in (1)
$2 x_1^3+\left(-6 x_1^3\right)+4=0 \Rightarrow 4 x_1^3=4 \Rightarrow x_1=1 \text {. Hence } b=-6$