A chemical engineer models the temperature gradient G(s)G(s)G(s) (in °C/cm) along a cooling fin, where s s\,s is the distance from the heat source in cm, as:
G(s)=as3−9s2+bs+14 G(s) = as^3 - 9s^2 + bs + 14 G(s)=as3−9s2+bs+14where a a\,a and b b\,b are constants.
When G(s)G(s)G(s) is divided by (s−4)(s - 4)(s−4), the remainder is 30.
Use the remainder theorem to show that
16a+b=40 16a + b = 40 16a+b=40Given also that (s−1)(s - 1)(s−1) is a factor of G(s)G(s)G(s),
find the value of a a\,a and the value of bbb.
Find G′(s)G'(s)G′(s).
Hence find the exact coordinates of the stationary points of the curve with equation y=G(s)y = G(s)y=G(s).
