Today, we will focus on the definitions of the terms that will appear in the
following section. Pay attention to the prepositions; they’re important and
you’re expected to understand the difference.
Definition 0.1. A function on an interval [a, b] is said to have:
• A global maximum at x = c if f (c) ≥ f (x) for all x ∈ [a, b]. We say that
f has a maximum of f (c) at c, and that f (c) is the maximum value of f .
• A global minimum at x = c if f (c) ≤ f (x) for all x ∈ [a, b]. We say that
f has a minimumof f (c) at c, and that f (c) is the maximum value of f .
• A local maximum at x = c if f (c) ≥ f (x) for all x “close to” c. We say
f has a local maximum of f (c) at x = c and that f (c) is a local maximum
of f .
• A local minimum at x = c if f (c) ≤ f (x) for all x “close to” c. We say
that f has a local minimum of f (c) at x = c and say that f (c) is a local
minimum of f .
Note that these definitions allow for ties.
For practice, let’s identify the local and global maxima and minima of the
picture in the geogebra file.
Notice where the local maxima and minima are. They occur at
• “hills” where the slope is zero.
• “corners” where the function fails to be differentiable.
• “Jumps” where the function fails to be continuous (and is therefore not
differentiable).
• Endpoints.
Definition 0.2. Let f be a function defined on [a, b]. We say that c is a critical
point of f if any one of the following conditions holds:
• c = a or c = b (These are the endpoints of the interval)
• f is not differentiable at a (“jump,”“corner,” or vertical tangent line.)
• f ′ (c) = 0 (flat)
The reason that critical points are important is that a local maximum or minimum of a function can only occur at a critical point!
Example 0.3. Find the critical points of f (x) = x3 − 12x on (−∞, ∞). Are
these critical points local maxima, local minima, or neither?
f ′ (x) = 3x2 − 12.
This is zero when x2 − 4 = 0. This happens when x = 2 or x = −2. From what
we know about cubic functions, f (x) is increasing when x is very negative and
when x is very positive, so there should be a local maximum at x = −2 and a
local minimum at x = 2.
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