Definitions [2]
A function may attain a maximum value at a point if its value there is greater than nearby values, and it may attain a minimum value if its value there is smaller than nearby values. These are often called extreme values.
A point in the domain of a function is called a critical point if either the derivative is zero there or the derivative does not exist there. Critical points are checked while locating possible maxima or minima.
Theorems and Laws [2]
Assume f'(c) = 0 and the second derivative exists at c:
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Local Maxima: f''(c) < 0
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Local Minima: f''(c) > 0
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Test Fails: f''(c) = 0. If this happens, you must go back and use the First Derivative Test to check if it is a maxima, minima, or point of inflection.
Let c be a critical point of a continuous function f:
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Local Maxima: f'(x) changes sign from positive to negative as x increases through c.
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Local Minima: f'(x) changes sign from negative to positive as x increases through c.
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Point of Inflection: f'(x) does not change sign as x passes through c (it is neither a maxima nor a minima).
Key Points
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Maxima and minima are extreme values of a function.
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Critical points occur where \(f'(x)=0\) or \(f'(x)\) is not defined.
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If \(f'(x)\) changes from positive to negative, the function has a local maximum.
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If \(f'(x)\) changes from negative to positive, the function has a local minimum.
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If \(f''(c) < 0\), there is a local maximum at \(x=c\).
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If \(f''(c) > 0\), there is a local minimum at \(x=c\).
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For absolute extrema on \([a,b]\), compare values at critical points and endpoints.
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Not every critical point gives a maximum or minimum.
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The second derivative test is quick, but the first derivative test is often more reliable in detailed reasoning.
