Question

a winding river is modeled br r(x)=x^3+4x^2+4x+12. bridges will be built along the x axis, as needed, to cross the river. how many bridges will need to be built?

Answer 1

The Winding river is modeled by r(x)= $x^{3}+ 4 x^{2}+4 x+12$

As bridges will be built along x-Axis to cross the river.

so,the bridges work as a tangent to river.

r'(x)=$3x^{2}+8 x+4$

So , r'(x)=0

→$3x^{2}+8 x+4$=0

→3x²+6 x+ 2 x+4=0

→3 x(x+2)+2(x+2)=0

→(x+2)(3 x+2)=0

x=-2, x= -2/3

As there are two values of x, from where we can draw tangents or bridges to the river.So there are maximum of two bridges that we  need to  built.