Engineers often rely on second-derivative information when deciding whether a…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Engineers often rely on second-derivative information when deciding whether a critical point of a cost or surface-area function represents an actual minimum in a design problem.
(a) Describe, in general terms, how the first and second derivatives of a differentiable surface-area function S(x) are used to locate and classify critical points.
(b) Explain why a positive value of S′′(x) at a critical point implies that the corresponding design uses the least possible amount of material (locally).
(c) Discuss one practical limitation of relying solely on second-derivative tests when optimizing real-world designs.