Concavity and the second derivative test? (aircraft glide path)?

Question

a small aircraft starts its descent from an altitude of one mile, 4 miles west of the runway.

A) find the cubic f(x)= ax^3+bx^3+cx+d on the interval [-4, 0] that describes a smooth glide path for the landing.

B) if the glide path of the plane is described byt the function in part (A), when would the plane be descending at the most rapid rate?

Answer 1

A) I think that a 'smooth' path is one that starts at a zero slope and ends on a zero slope. So, to get 4 eqns in 4 variables, (a,b,c,d)

set f(-4)=1 mile, f(0)= 0 mile

and

f(-4)=f'(0)=0.

That will give you a system of equations.

B) You should be able to get this part. Symmetry suggests that it will occur at x=-2.