Cubic Function Inflection Analysis
Consider the cubic function \(f(x) = ax^3 + bx^2 + cx + d\). Determine the x-coordinate of any potential inflection point by solving \(f”(x)=0\). Subsequently, perform a sign check on \(f”(x)\) around this x-coordinate to rigorously confirm whether it is indeed an inflection point.
Identifying Stationary Points in a Cubic Function
Consider the cubic function \(f(x) = 2x^3 – 9x^2 + 12x + 1\). (a) Differentiate \(f(x)\) with respect to \(x\) and factorize the derivative. (b) Find the values of \(x\) where \(f(x)\) has stationary points and use the first derivative test to classify these points as local maxima or minima.
Intersection Analysis: Cubic and Trigonometric Functions
Using a graphing calculator (GDC), analyze the intersections between the cubic function \(f(x) = x^3 – 3x^2 + 2x + 1\) and the trigonometric function \(g(x) = \sin(x)\). (a) Set up the equation that represents the intersection of these two functions. (b) Use your GDC to approximate the solutions of the equation. (c) Interpret the […]
Estimating Turning Points of a Cubic Function
Consider the function \(f(x) = x^3 – 3x\). Use a Graphical Display Calculator (GDC) to estimate the turning points of the graph. Identify each turning point as either a maximum or a minimum, and state their approximate coordinates.
Cubic Intercepts Factorization
Determine the x-intercepts of the cubic function \(f(x) = x^3 – 6x^2 + 11x – 6\) by factorisation.