The cost to produce a certain product depends on 3 main variables, denoted p, q, and r. If C(p, q, r) = C0e−2p + 7q − 9r represents the cost function, find the greatest rate of decrease of this function at the point P0(100, 55, 10).?

Answer:[tex]\sqrt{134}C_0e^{95}[/tex]Step-by-step explanation:We are given that the cost function is  given by [tex]C(p,q,r)=C_0e^{-2p+7q-9r}[/tex][tex]\nabla C(p,q,r)=< -2C_0e^{-2p+7q-9r},7C_0e^{-2p+7q-9r},-9C_0e^{-2p+7q-9r}>[/tex]We have to find the rate of decrease of this function at the point [tex]P_0(100,55,10)[/tex][tex]\nabla C(100,55,10)=<-2C_0e^{-200+385-90},7C_0e^{-200+385-90},-9C_0e^{-200+385-90}>[/tex][tex]\nabla C(100, 55,10)=<-2C_0e^{95},7C_0e^{95}, -9C_0e^{95}>[/tex][tex]\mid \nabla C(100, 55, 10)\mid=\sqrt{(-2C_0e^{95})^2+(7C_0e^{95})^2+(-9C_0e^{95})^2}[/tex]Greatest rate of decrease of this point at the point [tex]P_0(100,55,10)[/tex] is given by [tex]\mid \nabla C(100, 55, 10)\mid=\sqrt{134}C_0e^{95}[/tex]