MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Let h(x)=x4−4x3+6x2−3, defined for all real numbers.
(a) Compute h′′(x) and solve h′′(x)=0.
(b) Identify every real number where h′′(x) is zero or undefined. These are the only values where concavity might change. List them explicitly.
(c) Analyze the sign of h′′(x) on the intervals determined by the values in part (b). Clearly indicate the sign on each open interval.
(d) Based on your work above, state whether x=1 is an inflection point of the graph of h. Provide a complete justification, and list the coordinates of all inflection points of h or state that none exist.