For each of the following functions, determine the order m of the zero of the deriva- tive f' at zo and show explicitly that the function is not one-to-one in any neighbor- hood of zo. (a) f(z) = z² + 2z + 1, zo = −1 (b) f(z) = cos z, zo = 0, ±, ±2л,... (c) f(z) = e²³, zo = = 0

How would you differentiate this using the differentiation rules?

Consider the graph of f', the derivative of f, that is given below. Use it to answer the following questions. a.) Where does f(x) have a local maximum? Explain how you know. b.) Where does f(x) have a local minimum? -5- c.) Where is f concave up? Explain how you know. Answer in interval notation. d.) Which is larger, f(-2) or f(-1), or is there not enough information to tell? 8. Use Newton's method with the specified initial approximation x₁ to find x4, the fourth approximation to the root of the equation x³ - 4x² + 5 = 0, x₁ = 4. Show each approximation along the way. جتنا -5 -4 -3 -2 -1 2+ 1 0 -1+ 2 -4+ 2 f'(x) +X 4 5 6

The function, f(x), in the graph below has f(1) = 1 and f'(1) = = 2. Which of the following are the coordinates of points A and B? A f(x) 1 1.1 Select one: A(1,1) and B(1.1, 1.2) A(1,2) and B(1.1, 1.21) A(1.2,1) and B(1, 2) A(1,1) and B(1.1, 1.21) A(2,1) and B(1.2, 1.1)