This is about calculus ap. answer the following questions like tutoring me. Make a number line if needed and please answer it in steps. if you can can you write it instead of a write out typing. this focuses on. extreme values of functions, mean value theorem, connecting f' and f" with the graph of f, linearization and etc.

1. Using the function f(x) = x3 + x2 - 8x + 3 and analytic methods, find: | ONE a ) The intervals on which the function is increasing and decreasing. 3. The derivative of a function is f'(x) = (3x - 4)2(x + 5). Find: a) Local maximum 3x2 + 2x - 8 =0 (3x - 2) ( x14)= 0 - 10 - 4 0 21 * = 213 , - 4 b) The intervals on which the function is concave up and down. b) Local minimum 1 2 3 F ( x ) 1 12 f' ( x ) 0 -7 c) Extrema and their values (x, y). f" ( x ) -10 5. The table above shows the a twice differentiable function, f and specific values of the function. If A(x) = 2f(f(x)), does A have a critical value at x = 2? If so, is it a max, min, or neither? d) Inflection points and their values (x, y). c) Point of inflection 6. The derivative of a function is g'(x) = 2x2 - 4x + 3 and the graph of g passes through the point P(-2,4). Find the linearization of g at x = -2, and use it to estimate g(-2.1). Is the value an under or overestimate? 2 . Below is the graph of a function, f. Use the graph to estimate: . The graph of a continuous function's derivative g'(x) is shown below. Using the graph, estimate : 7. Let f be a differentiable function with f(1) = 14, f'(1) = -12, and f"(x) > 0 from (-1,3). - 4 Approximate the value of f(1.5) and state whether it's an overestimate or an underestimate. "a ) Where g (x ) is increasing and decreasing . Justify your answer . a) The extrema and where they occur. b) Where f' (x) > 0. b) Find all local extrema of g(x) and state whether they are a min or a max. Justify. c) Where f' (x) 0. c ) Find all points of inflection and justify your answers. e) Where f"(x)