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Math 165 - Assignment #4 Name: Due 7/16/2013 SHOW YOUR WORK! This assignment covers sections 1 - 3 from chapter 3. Make sure you justify your answer clearly! Simply writing the final result does not mean you receive full credit. You must also show you understand the procedure. 1.- Apply Theorem A on page 151 to determine whether the following functions attain a maximum value. Then, identify the critical points and find the global maximum value and minimum value of the given interval. (a) f (x) = x4 − 3x3 + 3 on the interval [−2, 2]. (b) f (x) = x 1+x2 on the interval [−1, 4]. 2.- Sketch the graph of a function with the given properties: f is continuous, but not necessarily differentiable, has domain [−3, 3], reaches a maximum of 3 (attained when x = 2) and a minimum of -3 (attained when x = −2). Additionally, f has no stationary points. f (x) x 3.- Consider the function f (x) = x3 − 6x2 + 9x + 2. (a) Use the Monotonicity Theorem on page 155 to find where f is increasing and where it is decreasing. (b) Use the Concavity Theorem on page 157 to find where f is concave up and where it is concave down. 4.- Consider the function f (x) = cos2 (x) on the interval (−π/2, π). Identity the critical points. (Hint: take a look at example 3 on page 164) (a) Use the First Derivative Test to decide which of the critical points five a local maximum and which give a local minimum. (b) Use the Second Derivative Test to decide which of the critical points five a local maximum and which give a local minimum 5.- The first derivative f 0 is given by f 0 (x) = x(A − x)(x − B) where 0 < A < B. Find all values of x such that the function f has a local minimum and all values of x such that it has a local maximum.