Maxima, Minima and the Second Derivative Test
Interior extrema occur at critical points, and the sign of the second derivative distinguishes a local maximum from a local minimum.
What is Maxima, Minima and the Second Derivative Test?
Interior extrema occur at critical points, and the sign of the second derivative distinguishes a local maximum from a local minimum.
Key formula / rule: Critical points are where f' is zero or fails to exist.
Key points
- Compute the local extrema of a function using derivative tests.
- Model a described optimisation problem and solve it with calculus.
Common exam trap
Reporting a critical point as an extremum without a test.
Definitions
- Term
Maxima, Minima and the Second Derivative Test
- Meaning
Interior extrema occur at critical points, and the sign of the second derivative distinguishes a local maximum from a local minimum.
- Term
Maxima, Minima and the Second Derivative Test — explanation
- Meaning
The first derivative locates candidates and the second classifies them; where the second derivative vanishes, the first-derivative sign change must be used instead. Closed-interval problems also require the endpoints to be tested.
Learning objectives
Compute the local extrema of a function using derivative tests.
Model a described optimisation problem and solve it with calculus.
Formulae
- Key point
Critical points are where f' is zero or fails to exist.
- Key point
f'' < 0 indicates a local maximum and f'' > 0 a local minimum.
- Key point
On a closed interval, absolute extrema may occur at the endpoints.
Prerequisites
MAT-U2-AD-T1-S1-C1
Common mistakes
Reporting a critical point as an extremum without a test.
Ignoring the endpoints when finding the absolute maximum on a closed interval.
Keywords
Maxima
Minima
Second
Derivative
Practice preview
Find the minimum value of f(x) = x^(2/3) + 2x^(1/3) in the interval [-1, 1].…
hard
According to the Second Derivative Test, if f'(c) = 0 and f''(c) < 0, then x = c is a point of:…
easy
Which of the following statements about the Second Derivative Test for local extrema is INCORRECT?…
medium