3. Differentiation

Optimisation — Quiz

Test your understanding of optimisation with 5 practice questions.

Practice Questions

Question 1

A rectangular field is to be fenced off with 400 meters of fencing. One side of the field is against an existing wall, so no fencing is needed for that side. If the length of the side parallel to the wall is $L$ and the lengths of the two sides perpendicular to the wall are $W$, what is the objective function for the area of the field in terms of $W$?

Question 2

A function is given by $f(x) = x^3 - 6x^2 + 9x + 10$. To find the critical points, the first derivative $f'(x)$ is set to zero. What are the critical points of this function?

Question 3

A company's profit function is $P(x) = -0.5x^2 + 20x - 150$, where $x$ is the number of units produced. To maximise profit, what is the optimal number of units to produce?

Question 4

A cylindrical can has a volume of $100\pi \text{ cm}^3$. The volume formula is $V = \pi r^2 h$. If the surface area is given by $A = 2\pi r^2 + 2\pi rh$, express the surface area $A$ as a function of the radius $r$ only.

Question 5

A farmer wants to enclose a rectangular area for his chickens. He has 120 meters of fencing. What is the maximum area he can enclose?