Local maxima
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This document discusses local maxima, local minima, and inflection points of functions. It provides the following key points: - A function f has a local minimum at p if f(p) is less than or equal to f(x) in a small interval around p. It has a local maximum if f(p) is greater than or equal to f(x) in that interval. It has an inflection point if the concavity of f changes at p. - Table 1 shows that if f'(p) = 0 and f''(p) is positive, f has a local minimum at p. If f'(p) = 0 and f''(p) is negative
![Local Maxima, Local Minima, and Inflection Points
Let f be a function defined on an interval [a,b] or (a,b), and let p be a point in (a,b), i.e.,
not an endpoint, if the interval is closed.
• f has a local minimum at p if f(p) ≤ f(x) for all x in a small interval around p.
• f has a local maximum at p if f(p) ≥ f(x) for all x in a small interval around p.
• f has an inflection point at p if the concavity of f changes at p, i.e. if f is concave
down on one side of p and concave up on another.
We assume that f '(p) = 0 is only at isolated points — not everywhere on some interval.
This makes things simpler, as then the three terms defined above are mutually exclusive.
The results in the tables below require that f is differentiable at p, and possibly in some
small interval around p. Some of them require that f be twice differentiable.
Table 1: Information about f at p from
the first and second derivatives at p
f '(p) f ''(p) At p, f has a_____ Examples
0 positive local minimum f(x) = x2, p = 0.
0 negative local maximum f(x) = 1− x2, p = 0.
local minimum, local f(x) = x4, p = 0. [min]
0 0 maximum, or inflection f(x) = 1− x4, p = 0. [max]
point f(x) = x3, p = 0. [inf pt]
f(x) = tan(x), p = 0. [yes]
nonzero 0 possible inflection point
f(x) = x4 + x, p = 0. [no]
nonzero nonzero none of the above
In the ambiguous cases above, we may look at the higher derivatives. For example, if
f '(p) = f ''(p) = 0, then
• If f (3)(p) ≠ 0, then f has an inflection point at p.
• Otherwise, if f (4)(p) ≠ 0, then f has a local minimum at p if f (4)(p) > 0 and a local
maximum if f (4)(p) < 0.](https://image.slidesharecdn.com/localmaxima-121121234241-phpapp02/85/Local-maxima-1-320.jpg)
Local maxima
- 1. Local Maxima, Local Minima, and Inflection Points Let f be a function defined on an interval [a,b] or (a,b), and let p be a point in (a,b), i.e., not an endpoint, if the interval is closed. • f has a local minimum at p if f(p) ≤ f(x) for all x in a small interval around p. • f has a local maximum at p if f(p) ≥ f(x) for all x in a small interval around p. • f has an inflection point at p if the concavity of f changes at p, i.e. if f is concave down on one side of p and concave up on another. We assume that f '(p) = 0 is only at isolated points — not everywhere on some interval. This makes things simpler, as then the three terms defined above are mutually exclusive. The results in the tables below require that f is differentiable at p, and possibly in some small interval around p. Some of them require that f be twice differentiable. Table 1: Information about f at p from the first and second derivatives at p f '(p) f ''(p) At p, f has a_____ Examples 0 positive local minimum f(x) = x2, p = 0. 0 negative local maximum f(x) = 1− x2, p = 0. local minimum, local f(x) = x4, p = 0. [min] 0 0 maximum, or inflection f(x) = 1− x4, p = 0. [max] point f(x) = x3, p = 0. [inf pt] f(x) = tan(x), p = 0. [yes] nonzero 0 possible inflection point f(x) = x4 + x, p = 0. [no] nonzero nonzero none of the above In the ambiguous cases above, we may look at the higher derivatives. For example, if f '(p) = f ''(p) = 0, then • If f (3)(p) ≠ 0, then f has an inflection point at p. • Otherwise, if f (4)(p) ≠ 0, then f has a local minimum at p if f (4)(p) > 0 and a local maximum if f (4)(p) < 0.
- 2. An alternative is to look at the first (and possibly second) derivative of f in some small interval around p. This interval may be as small as we wish, as long as its size is greater than 0. Table 2: Information about f at p from the first and second derivatives in a small interval around p Change in f '(x) as x moves f '(p) At p, f has a_____ from left to right of p f '(x) changes from negative 0 Local minimum to positive at p f '(x) changes from positive to 0 Local maximum negative at p f '(x) has the same sign on 0 both sides of p. Inflection point (Implies f ''(p) = 0.) Change in f ''(x) as x moves f '(p) f ''(p) At p, f has a_____ from left to right of p nonzero 0 f ''(x) changes sign at p. Inflection point f ''(x) has the same sign on None of the above. nonzero 0 (However, f ' has an both sides of p. inflection point at p.)