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Avon High School AP Calculus AB Name ______________________________________ Period _____ Score ______ / 10 HW 3.1 – Extrema on a Closed Interval. Find the absolute extrema of the given function on each indicated interval. 1.) f ( x) x3 3x 1 2.) f ( x) x 4 8x 2 2 a.) on 0, 2 a.) on 3,1 b.) on 3, 2 b.) on 1,3 3.) f ( x) x 4.) f ( x) sin x cos x 2 3 a.) on 4, 2 a.) on 0, 2 b.) on 1,3 b.) on , 2 5.) f ( x) 6.) f ( x) 2 x x 2 3x 2 x 3 a.) on 2, 2 a.) on 0,1 b.) on 2,8 b.) on 3, 4 7.) Determine whether each labeled point is an absolute maximum or minimum, a relative maximum or minimum or none. A E B F C G D 8.) Determine whether each statement is True of False. If a statement is false, explain why or give an example that shows it to be false. a.) The maximum of a function that is continuous on a closed interval can occur at two different values in the interval. b.) If a function is continuous on a closed interval, then it must have a minimum on the interval c.) If x = c is a critical number of the function f, then it is also a critical number of the function g ( x) f ( x) k , where k is a constant. d.) If x = c is a critical number of the function f, then it is also a critical number of the function g ( x) f ( x k ) , where k is a constant. e.) Let the function f be differentiable on an interval I containing c. If f has a maximum value at x = c, then f has a minimum value at x = c. f.) A cubic function defined by f ( x) ax3 bx 2 cx d where a 0 will always have exactly three critical numbers. 9.) What is the smallest possible slope for y x3 3x 2 5x 1 ? 10.) If a particle moves along a straight line according to s(t ) t 4 4t 3 6t 2 20 , find a.) the maximum and minimum velocity on 0 t 3. b.) the maximum and minimum acceleration on 0 t 3.