Free Response- Suppose that the derivative of a function, f(x), is given by the equation f"(x) = -2x(x + 2)²(x – 3). Additionally, f(-1) = -2 and the graph of («x) is concave up at x = -1. a. Perform a sign analysis on f' and determine at which value(s) of x, if any, that the graph of f would have a horizontal tangent but not a relative maximum or minimum. Justify your answer. b. On what interval(s) of x is the graph of f(x) increasing? Decreasing? Justify your answer. c. What is the tangent line approximation of the value of f(-0.9)? Show your work.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter1: Functions
Section1.2: Functions Given By Tables
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I need help for a, b, c, and d please

Free Response-
Suppose that the derivative of a function, f(x), is given by the equation f'(x) = -2x(x + 2)°(x – 3).
Additionally, f(-1) = -2 and the graph of (x) is concave up at x = -1.
a. Perform a sign analysis on f' and determine at which value(s) of x, if any, that the graph of f would have a
horizontal tangent but not a relative maximum or minimum. Justify your answer.
b. On what interval(s) of x is the graph of f(x) increasing? Decreasing? Justify your answer.
c. What is the tangent line approximation of the value of f(-0.9)? Show your work.
d. Will this approximation found in part c be less than or greater than the actual value of f(-0.9)? Justify your
answer.
Transcribed Image Text:Free Response- Suppose that the derivative of a function, f(x), is given by the equation f'(x) = -2x(x + 2)°(x – 3). Additionally, f(-1) = -2 and the graph of (x) is concave up at x = -1. a. Perform a sign analysis on f' and determine at which value(s) of x, if any, that the graph of f would have a horizontal tangent but not a relative maximum or minimum. Justify your answer. b. On what interval(s) of x is the graph of f(x) increasing? Decreasing? Justify your answer. c. What is the tangent line approximation of the value of f(-0.9)? Show your work. d. Will this approximation found in part c be less than or greater than the actual value of f(-0.9)? Justify your answer.
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