Stationary Points And Optimisation
Bayani Gaba-gaba
What is the tallest box you can make from a sheet of card? At what speed does a car use the least fuel? These are optimisation problems, and calculus gives you a precise, systematic way to solve them. By finding the points where a curve levels out, you can identify its highest and lowest values and use that information to design the best solution.
You will learn to locate stationary points by setting the first derivative equal to zero, classify them as maxima or minima using the second derivative, and apply these techniques to practical problems where a quantity must be maximised or minimised subject to a constraint.
Manufura
- Find and classify stationary points and turning points
- Solve maxima and minima problems, including in the context of practical problems
- Justify whether a stationary point is a maximum or minimum
Takardar Darasi
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- At a stationary point, which of the following is true? A) dy/dx > 0 B) dy/dx < 0 C) dy/dx = 0 D) d2y/dx2 = 0 Answer: C
- If d2y/dx2 > 0 at a stationary point, the point is: A) A maximum B) A minimum C) A point of inflection D) Cannot be determined Answer: B
- The curve y = x^3 - 3x has stationary points at: A) x = 0 only B) x = 1 and x = -1 C) x = 3 and x = -3 D) x = 0 and x = 3 Answer: B
- A rectangle has a fixed perimeter of 24 cm. The maximum area is: A) 24 cm squared B) 36 cm squared C) 48 cm squared D) 144 cm squared Answer: B
- At a point of inflection where dy/dx = 0, which is true? A) d2y/dx2 > 0 B) d2y/dx2 < 0 C) The sign of dy/dx does not change D) The curve has a vertical tangent Answer: C
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