Use the first derivative of a function to determine and identify the nature of stationary points.

A manufacturer models the daily profit, \(P\) (in dollars), from producing \(x\) units of a product using the function \[ P(x) = -2x^3 + 45x^2 - 180x + 500, \quad 0 \leq x \leq 20 \] where \(x\) is the number of units produced per day. (a) Determine the values of \(x\) at which stationary points occur. [2 marks] (b) Use the first derivative to identify the nature of each stationary point found in part (a). [2 marks]

A pharmaceutical company models the concentration of a drug in the bloodstream using the function \( C(t) = 15te^{-0.2t} \), where \( C \) is the concentration in milligrams per litre and \( t \) is the time in hours after administration, \( t \geq 0 \). (a) Use the product rule to find \( C'(t) \). (1 mark) (b) Determine the time at which a stationary point occurs. (1 mark) (c) Use the first derivative to identify the nature of this stationary point. (1 mark) (d) State the maximum drug concentration in the bloodstream, correct to two decimal places. (1 mark)

A container is being filled with water. The volume of water, \(V\) litres, in the container at time \(t\) minutes is modelled by the function \[ V(t) = 120t - 15t^2 + t^3, \quad 0 \leq t \leq 10 \] (a) Determine the rate at which the volume is changing at \(t = 2\) minutes. (1 mark) (b) Find the times at which the volume has stationary points. (2 marks) (c) Use the first derivative to determine the nature of each stationary point and interpret what each represents in the context of the container filling. (2 marks)