${\text{If the slope of line through the origin is tangent to the curve y}}^{ }=x^3+x+16 \text{is m ,then the value of m is}$

$y^{ }=x^3+x+16$

${\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}=3x_1^2+1$

$3x_1^2+1=\frac{y_1}{x_1}$$=slope of the line pas\sin g through \left(0,0\right) and \left(x_1,y_1\right)$

$\Rightarrow 3x_1^3+x_1=x_1^3+x_1+16$

$\Rightarrow 2x_1^3=16$

$\Rightarrow x_1=2$$\Rightarrow y_1=26$

 $\Rightarrow$$m=13$