Q4 A function y = f(t) is defined and continuous for all real numbers t on the closed interval [-2.5, 6.5]. The first derivative, y' = f'(t) is sketched below. For each question, give a short and brief explanation. -2 -1 2 ЛУ -1- -2- 1 2 3 f'(t) 4 (a) At what value(s) of t does the function f has critical point(s)? (b) On what open interval(s), the function f is decreasing. (c) At what t value(s) does the function f has local minimum point(s)? 5 6
Q4 A function y = f(t) is defined and continuous for all real numbers t on the closed interval [-2.5, 6.5]. The first derivative, y' = f'(t) is sketched below. For each question, give a short and brief explanation. -2 -1 2 ЛУ -1- -2- 1 2 3 f'(t) 4 (a) At what value(s) of t does the function f has critical point(s)? (b) On what open interval(s), the function f is decreasing. (c) At what t value(s) does the function f has local minimum point(s)? 5 6
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Question 4**
A function \( y = f(t) \) is defined and continuous for all real numbers \( t \) on the closed interval \([-2.5, 6.5]\). The first derivative, \( y' = f'(t) \), is sketched below. For each question, give a short and brief explanation.
![Graph Image]
The graph displayed is of the first derivative \( f'(t) \) with respect to \( t \). The horizontal axis (\( t \)-axis) ranges from \(-2\) to \( 6\) and the vertical axis (\( y \)-axis) ranges from \(-2\) to \( 2\).
In the graph, \( f'(t) \) starts from \( -2 \) on the \( y \)-axis when \( t = -2 \), reaches a peak of roughly \( 2 \) around \( t \approx 1\), goes down and crosses the \( t \)-axis at \( t \approx 3 \), dips to a low point slightly above \( -1 \) around \( t \approx 4 \), and then rises again.
**Questions:**
(a) At what value(s) of \( t \) does the function \( f \) have critical point(s)?
(b) On what open interval(s), is the function \( f \) decreasing?
(c) At what \( t \) value(s) does the function \( f \) have local minimum point(s)?
(d) On what open interval does the graph of \( f' \) concave down on?
(e) At what \( t \) values does the graph of \( f \) have inflection point(s)?
**Answers:**
(a) Critical points occur where \( f'(t) = 0 \). From the graph, the function \( f \) has critical points at \( t \approx 3 \) and \( t \approx 5 \).
(b) The function \( f \) is decreasing where \( f'(t) < 0 \). From the graph, \( f \) is decreasing on the intervals \( (-2.5, -1.5) \) and \( (3, 4.5) \).
(c) Local minimum points occur where \( f'(t) = 0 \) and changes from negative to positive. From the graph, this occurs at \( t \approx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2cc04b1e-803c-4d9d-b550-d2ae04bcc2b3%2Fdd0b6a00-262e-4946-9bf7-eb594e913898%2Fcsdo4rd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 4**
A function \( y = f(t) \) is defined and continuous for all real numbers \( t \) on the closed interval \([-2.5, 6.5]\). The first derivative, \( y' = f'(t) \), is sketched below. For each question, give a short and brief explanation.
![Graph Image]
The graph displayed is of the first derivative \( f'(t) \) with respect to \( t \). The horizontal axis (\( t \)-axis) ranges from \(-2\) to \( 6\) and the vertical axis (\( y \)-axis) ranges from \(-2\) to \( 2\).
In the graph, \( f'(t) \) starts from \( -2 \) on the \( y \)-axis when \( t = -2 \), reaches a peak of roughly \( 2 \) around \( t \approx 1\), goes down and crosses the \( t \)-axis at \( t \approx 3 \), dips to a low point slightly above \( -1 \) around \( t \approx 4 \), and then rises again.
**Questions:**
(a) At what value(s) of \( t \) does the function \( f \) have critical point(s)?
(b) On what open interval(s), is the function \( f \) decreasing?
(c) At what \( t \) value(s) does the function \( f \) have local minimum point(s)?
(d) On what open interval does the graph of \( f' \) concave down on?
(e) At what \( t \) values does the graph of \( f \) have inflection point(s)?
**Answers:**
(a) Critical points occur where \( f'(t) = 0 \). From the graph, the function \( f \) has critical points at \( t \approx 3 \) and \( t \approx 5 \).
(b) The function \( f \) is decreasing where \( f'(t) < 0 \). From the graph, \( f \) is decreasing on the intervals \( (-2.5, -1.5) \) and \( (3, 4.5) \).
(c) Local minimum points occur where \( f'(t) = 0 \) and changes from negative to positive. From the graph, this occurs at \( t \approx
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