A Cubic Function with 4 Unknowns & 3 Inverse Functions | Can You Find Them? #CentroidX | Lecture 47

In this lecture, we solve two complete exercises involving cubic functions, monotonicity, inverse functions, graphs, and algebraic determination of inverses. 📌 Exercise 1 — Cubic Function & Three Inverse Functions We begin with a cubic function containing four unknown parameters A, B, C, and D. Using specific points on the curve together with derivative information at its local maximum and local minimum, we determine all four parameters. We then study the function on three different intervals by analyzing its continuity and monotonicity. This allows us to prove that the function is invertible on each interval and therefore admits three inverse functions. For one of these inverse functions, we also determine the point of intersection between the inverse curve and the original cubic curve, using the relationship between a function and its inverse. 📌 Exercise 2 — Irrational Function & Its Inverse In the second exercise, we study an irrational function: Draw and analyze its graph Construct the graph of its inverse function Use the symmetry between a function and its inverse Determine the algebraic expression of the inverse function This lecture connects several fundamental ideas: cubic functions, derivatives, local extrema, monotonicity, continuity, inverse functions, graph transformations, and algebraic problem solving. 📚 CentroidX — Mathematics through rigorous reasoning, graphical interpretation, and problem solving. #CentroidX #Mathematics #CubicFunction #InverseFunctions #Functions #Monotonicity #IrrationalFunctions #MathExercises