Answer to Question #11985 in Functional Analysis for lee
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?
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.
since f(x)=ax^3+b , a≠0. a)for all values of b, 1. when a positive => the graph falls to the left and rises to the right. 2. when a negative => the graph falls to the right and rises to the left. point of inflexion : f''(x)=0, 6ax=0 => x=0 and y=f(x)=b hence point of inflexion is: (0,b). for all values of b.