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November 14, 2013 Section 3.1 Extrema on an Interval Definition of Extrema Let f be defined on an interval I containing c. 1. f(c) is the minimum of f on I if f(c) ≤ f(x) for all x in I. 2. f(c) is the maximum of f on I if f(c) ≥ f(x) for all x in I. A min or max of f on I is called the extremum. Extreme Value Theorem If f is continuous on [a, b], then f has both a min and max on [a, b]. Definition of Relative Extrema 1. If there is an open interval containing c on which f(c) is a maximum, then f(c) is called a relative maximum of f, or we say "f has a relative max at 2. If there is an open interval containing c on which f(c) is a minimum, then f(c) is called a relative minimum of f, or we say "f has a relative min at 1 November 14, 2013 Definition of a Critical Number Let f be defined at c. If f '(c) = 0 or if f is not differentiable at c, then c is a critical number of f. Theorem 3.2 If f has a relative min or max at x = c, then c is a critical number of f. Guidelines to find the extrema of a continuous function f on [a, b] 1. Find the critical numbers of f in (a, b), 2. Evaluate f at each critical number in (a, b), 3. Evaluate f at each endpoint of [a, b], 4. The least of these values in the min. The greatest is the max. 2 November 14, 2013 Section 3.2 Rolle's Theorem and the Mean Value Theorem Rolle's Theorem Let f be continuous on [a, b] and differentiable on (a, b). If f(a) = f(b), then there is at least one number c in (a, b) such that f '(c) = 0. f(b) = f(a) a c b The Mean Value Theorem If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c in (a, b) such that f(b) − f(a) f '(c) = b−a f(b) f(a) a b 3 November 14, 2013 4 November 14, 2013 Section 3.3 Increasing and Decreasing Functions and the First Derivative Test Definitions of Increasing and Decreasing Functions A function f is increasing on an interval if for any two numbers a and b in the interval, a < b implies f(a) < f(b). A function f is decreasing on an interval if for any two numbers a and b in the interval, a < b implies f(a) > f(b). f is increasing on (a, b) • f(b) f(a) • a f (a) b f is decreasing on (a, b) • • f (b) a b 5 November 14, 2013 Test for Increasing and Decreasing Functions Let f be a function that is continuous on [a, b] and differentiable on (a, b). 1. If f '(x) > 0 for all x in (a, b), f is increasing on [a, b]. 2. If f '(x) < 0 for all x in (a, b), f is decreasing on [a, b]. 3. If f '(x) = 0 for all x in (a, b), f is constant on [a, b]. The First Derivative Test Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows. 1. If f '(x) changes from negative to positive at c, then f has a relative minimum at (c, f(c)). 2. If f '(x) changes from positive to negative at c, then f has a relative maximum at (c, f(c)). 3. If f '(x) is positive on both sides of c or negative on both sides of c, then f(c) is neither a relative minimum nor a relative maximum. 6 November 14, 2013 Section 3.4 Concavity and the Second Derivative Test Definition of Concavity Let f be differentiable on an open interval I. The graph of f is concave upward on I if f ' is increasing on the interval and concave downward on I if f ' is decreasing on the interval. Test for Concavity Let f be a function whose second derivative exists on an open interval I. 1. If f "(x) > 0 for all x in I, then the graph of f is concave upward in I. 2. If f "(x) < 0 for all x in I, then the graph of f is concave downward in I. y = f (x) 7 November 14, 2013 Definition of Point of Inflection Let f be function that is continuous on an open interval and let c be a number in this interval. If the graph of f has a tangent line at this point (c, f (c)), then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward (or downward to upward) at the point. Points of Inflection If (c, f (c)) is a point of inflection of the graph of f, then either f "(c) = 0 or f " does not exist at x = c. The Second Derivative Test Let f be a function such that f '(c) = 0 and the second derivative of f exists on an open interval containing c. 1. If f "(c) > 0, then f has a relative min at (c, f (c)). 2. If f "(c) < 0, then f has a relative max at (c, f (c)). If f "(c) = 0, the test fails. Revert to using the First Derivative Test in this case. 8 November 14, 2013 6. Find values of a, b, c, and d so that f (x) = ax3 + bx2 + cx + d has an inflection point at (1, -1) and a relative extremum at (0, 3). 9