Part b
(i) Show that the surface area \(S(x)\) of the open-top container can be expressed in terms of \(x\) as \(S(x) = x^2 + \frac{128}{x}\). (ii) Find the value of \(x\) that minimizes the surface area \(S(x)\). Justify that this value corresponds to a minimum using derivative analysis, including the second derivative test.
Optimisation of a Container’s Dimensions
A manufacturer needs to design a rectangular container with an open top. The container must have a volume of \(32\) cubic units and a square base to minimize material costs. Let \(x\) be the side length of the square base and \(h\) be the height of the container, where \(h = \frac{32}{x^2}\).
Analysis of Stationary Points in a Composite Function
Consider the function \(f(x) = x^3 – 6x^2 + 9x + 2\) defined on the closed interval \([0, 4]\). Analyze the function\’s stationary points and endpoints to find the maximum and minimum values of the function on this interval. (a) Solve \(f'(x)=0\). (b) Evaluate the function at the endpoints \(x=0\) and \(x=4\). (c) In a […]
Composite and Inverse Functions: Domain Analysis
Consider the function \(f(x)= x^3 – 3x + 2\). (a) Prove that \(f(x)\) is not injective by demonstrating the existence of distinct values \(x_1\) and \(x_2\) for which \(f(x_1)=f(x_2)\). (b) Propose a restriction on the domain of \(f(x)\) that renders it invertible and justify your choice based on the function’s monotonicity.
Part b
Determine a domain restriction for \(f(x)=x^3-3x+2\) that ensures the function is invertible. Justify your choice using the function’s monotonicity properties.