GDC Assisted Optimisation: Finding Critical Points
Use your Graphical Display Calculator (GDC) to find the critical point of the function \(f(x) = x^2 – 6x + 10\). Explain how the GDC aids in computing the derivative and what the critical point indicates in the context of optimisation.
Concavity and Optimisation Analysis
Given the function \(f(x) = -x^2 + 4x + 1\), determine the nature of the critical point at \(x = 2\) using the second derivative test.
Optimisation of a Piecewise-Defined Function on a Restricted Domain
Consider the piecewise function \(f(x)\) defined by: \[ f(x) = \begin{cases} x^2 – 4 & \text{if } x < 2, \\ 8 – x & \text{if } x \geq 2 \end{cases} \] and let the domain be \([-3, 5]\). Determine the absolute minimum and absolute maximum values of \(f(x)\) on the given interval. Clearly state […]
Quotient Rule in Optimisation Context
Consider the function \(f(x)=\frac{x^2+1}{x-2}\). Using the quotient rule, find \(f'(x)\) and determine the critical point corresponding to a local extremum for \(x > 2\).
Understanding the Second Derivative Test in Optimisation
Consider a function \(f(x)\) that has a stationary point at \(x = c\). Explain how the second derivative test, using \(f”(c)\), helps to determine whether this stationary point is a local maximum, local minimum, or if the test is inconclusive.