Hello there, assistance would be terrific, thank you very much.

Question

Consider the function f(x)= x^3 + 2x^2 + bx.

a) The equation of the tangent line to the graph of this function at x = 1 is given by
y = ?

b) The tangent line intersects the x-axis at x = ?

c) For what value(s) of b is there no intersection? Why not?

Answer 1

y = x^3+2x^2+bx
y' = 3x^2+4x+b
y'(1) = 3+4+b = 7+b
y(1) = 1+2+b = 3+b

So the line through (1,3+b) with slope 7+b is

y-(3+b) = (7+b)(x-1)
y = (7+b)x - 4
y=0 at x = 4/(7+b)

So, under what condition will the tangent line(s) not intersect the x-axis?

See an example for b = -3:
http://www.wolframalpha.com/input/?i=plot+y%3Dx%5E3%2B2x%5E2-3x,+y%3D4x-4

Answer 2

Great, thank you Steve!