Ap calculus student handout about Concavity, inflection points. Please **step by step and use a **number line as needed to answers the questions. if you could answer the questions in papers instead.

AP CALCULUS Concavity and Inflection Justifying Inflection Points AP CALCULUS STUDENT HANDOUT Question 1 Suppose that f is differentiable and that f"(x) = (x - 1)? (x - 2)'. Which x-values have f"(x) =0? What are the first coordinates of any inflection points of ?? Finding Inflection Points Some values of a twice differentiable function, f (x) , and its first and second derivatives, f'(x) and f"(x) respectively, are given in the table below. For example, f'(3) - -2. Use the table to answer the questions that follow. Answer the questions in the scenarios below, then compare your answer with those of your group members and discuss any differences. Scenario 1: Suppose that f (x) = x* + x3 -3x2 7XX A( x ) 4 2 ON A Question 2 . Find the first and second derivatives of f (x). Suppose that g is differentiable. A graph of the derivative of g, that is, y = g'(x), is displayed below. Use - 2 that graph to answer these questions: which x-values have g"(x) = 0, and what are the first coordinates MX f. ( x ) 2 - 3 of any inflection points of g(x) ? 1. Does A(x) = (f (x))? have a critical point at x = 4? If A(x) does have a critical point, can you b. Does f(x) have any inflection points? If it does, find their coordinates and explain why they are determine whether it is a local maximum, local minimum, or neither? Explain your answer. inflection points. Graph of y = g'(x), the derivative of g(x) This graph has a horizontal tangent at x = -2.) Scenario 2: Suppose that g(x) = 2 x2 + x+1 . Then the first derivative, g'(x), is " +2x3 +3x2 -4x -2 ( x2 + *+ 1) 2 2. Does B(x) = f (x2) have a critical point at x = 2? If B(x) does have a critical point, can you 18x(x +1) and the second derivative, g"(x) , is ( x2 + x + 1)3 - ma . Does g (x) have any inflection points? If it does, find determine whether it is a local maximum, local minimum, or neither? Explain your answer. their coordinates and explain why they are inflection points. CollegeBoard 2017 College Board AP CALCULUS STUDENT HANDOUT 3. Does C(x) = f(f(x)) have a critical point at x = 3? if C(x) does have a critical point, can you determine whether it is a local maximum, local minimum, or neither? Explain your answer. Scenario 3: This is a graph of the derivative of h(x), which is a function defined and continuously differentiable on the interval [ A,G] . Use this graph of y = h'(x) to answer the following questions. AP CALCULUS STUDENT HANDOUT Check your understanding pose you are given an analytical representation (a formula) for a function f(x) . What steps could you use to identify the inflection points of the function? 4. Does D(x) = f(4x -2) have a critical point at x = 1? if D(x) does have a critical point, can you determine whether it is a local maximum, local minimum, or neither? Explain your answer. X=AX=B X=C X=Di " Suppose you are given a graphical representation for f'(x) , the derivative of a function f (x). How XEE X= G could you identify the inflection points of the function? The graph of y = hi(x) a. What are the x-coordinates of the inflection points of h(x) ? 5. Does E(x) = f (x+3) have a critical point at x = 0? if E(x) does have a critical point, can you determine whether it is a local maximum, local minimum, or neither? Explain your answer. b. Justify why those x-values are inflection points. CollegeBoard 2017 College Board