Term
| absolute (global) maximum |
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Definition
function value f(c) ∋
f(c) ≤ f(x) ∀x ∈ Df |
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| absolute (global) minimum |
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Definition
function value f(c) ∋
f(c) ≤ f(x) ∀x ∈ Df |
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Term
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Definition
| If f is continuous on a closed interval, the f attains both an absolute maximum & minimum on that interval. |
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Term
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Definition
function value f(c) ∋
f(c) ≥ f(x) ∀x
in some open interval containing c, an interior point of Df |
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Term
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Definition
function value f(c) ∋
f(c) ≤ f(x) ∀x
in some open interval containing c, an interior point of Df |
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Term
| First Derivative Theorem for Local Extreme Values |
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Definition
if f
in some open interval containing c has a local maximum/minimum value at an interior point c of its domain, and if f ' is defined at c, then:
f'(c) = 0 |
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Term
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Definition
| an interior point of the domain of f where f ' is 0 or undefined. |
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Term
| increasing on an interval |
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Definition
| if ∀x1, x2 in the interval with x1 < x2 and f(x1) < f(x2) |
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Term
| decreasing on an interval |
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Definition
| if ∀x1, x2 in the interval with x1 < x2 and f(x1) > f(x2). |
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Term
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Definition
| if the function is increasing or decreasing on that interval |
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Term
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Definition
Suppose that c is a critical point of a continuous function of f, and that f is differentiable at every point in some interval containing c, except possibly at c itself.
1) if f' changes from - to + at c, then f has a local minimum at c.
2) if f' changes from + to - at c, then f has a local maximum at c. |
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Term
| Intermediate Value Theorem |
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Definition
| A function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b). |
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Term
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Definition
| Suppose that y = f(x) is continous at every point of the closed interval [a,b] and differentiable at every point of its interior (a,b). If f(a) = f(b), then there is at least one number c ∈ (a,b) at which f'(c) = 0. |
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Term
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Definition
Suppose y = f(x) is continuous on a closed interval [a,b] and differentiable on the interval's interior (a,b). Then there is at least one point c ∈ (a,b) at which
f(b) - f(a) = f'(c)
b - a |
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Term
| concave up on an interval I if |
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Definition
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Term
| concave down on an interval I if |
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Definition
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Term
| Second Derivative Test for Concavity |
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Definition
Suppose that f" exists on an interval I.
1) if f"(x) > 0 on I, then f is concave up on I.
2) if f"(x) < 0 on I, then f is concave down on I. |
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Term
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Definition
| point on the domain where the function changes concavity. |
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Term
| Second Derivative Test for Local Extrema |
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Definition
Suppose f" is continuous on an open interval that contains x = c.
1) if f'(c) = 0 and f"(c) < 0, then f has a local maximum at x = c.
2) if f'(c) = 0 and f"(c) > 0, then f has a local minimum at x = c.
3) if f'(c) = 0, and f"(c) = 0, then the test fails. |
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