Answer to Question #11985 in Functional Analysis for lee

Question

consider a cubic function f:x-ax^3+b,where a and b are real numbers and a not equal to 0.the shape of the cubic function depends on the value of a and b.
a) By using any sitable tools.investigate the shape of the graph if both a and b have the same sign and if a and b have different sign. identify the point of inflexion in eash case.
b) Investigate the point of intersection of the graph of f and its tangent.What can you say about the number of point of intersection?

Answer 1

Suppose that& a≠0 (otherwise )given function would be constant).
a) Depending on a, if a>0 the graph falls to the left and rises to the right.
if a<0 the graph falls to the right and rises to the left.
point of inflexion : f''(x)=0,
& 6ax=0 therefore we get point (0,b)

b) A Tangent Line is a line which locally touches a curve at one and only one point.