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Unformatted text preview: Math 006 (Lecture 24) Increasing and Decreasing Functions Definition 1. Suppose y = f ( x ). 1. f ( x ) is increasing on the interval, a < x < b , if for any a < x 1 < x 2 < b , f ( x 1 ) ≤ f ( x 2 ). 2. f ( x ) is decreasing on the interval, a < x < b , if for any a < x 1 < x 2 < b , f ( x 2 ) ≤ f ( x 1 ). Example 1. Determine the interval for which the function f ( x ) = x 2 is increasing. Theorem 1. For the interval a < x < b , 1. f ( x ) is an increasing function on a < x < b if and only if f ( x ) > on a < x < b . 2. f ( x ) is an decreasing function on a < x < b if and only if f ( x ) < on a < x < b . Example 2. Given the function f ( x ) = 8 x- x 2 , (a) Which values of x correspond to horizontal tangent line? (b) For which values of x is f ( x ) increasing? Decreasing? 1 Critical Values Definition 2. The values of x in the domain of f where 1. f ( x ) = 0, or 2. f ( x ) does not exist are called the critical values of f ....
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- Fall '09