. D11: I can use derivatives to solve optimization problems. (a) Part of the graph of f(x) = x + 2x is given below. i. Does the graph indicate that x = 0 is a critical point? Why/Why not? ii. Use calculus and algebra to find all possible x values where f could possibly change direction (change from increase to decrease or vice versa). Does your algebra indicate that x = 0 is a critical point? Why/Why not? lasitos -2 0 -2 (b) Use the first derivative test to show that f does not have a maximum or a minimum at x = 0. (Hint: you may need to look up this test. Show why it is ok to apply the test to this situation.)
. D11: I can use derivatives to solve optimization problems. (a) Part of the graph of f(x) = x + 2x is given below. i. Does the graph indicate that x = 0 is a critical point? Why/Why not? ii. Use calculus and algebra to find all possible x values where f could possibly change direction (change from increase to decrease or vice versa). Does your algebra indicate that x = 0 is a critical point? Why/Why not? lasitos -2 0 -2 (b) Use the first derivative test to show that f does not have a maximum or a minimum at x = 0. (Hint: you may need to look up this test. Show why it is ok to apply the test to this situation.)
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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![**4. D11: I can use derivatives to solve optimization problems.**
(a) Part of the graph of \( f(x) = x + 2 \sqrt[3]{x} \) is given below.
i. Does the graph indicate that \( x = 0 \) is a critical point? Why/Why not?
ii. Use **calculus and algebra** to find all possible \( x \) values where \( f \) could possibly change direction (change from increase to decrease or vice versa). Does your algebra indicate that \( x = 0 \) is a critical point? Why/Why not?
**Graph:**
The graph shows a curve starting from the third quadrant, smoothly passing through the origin (0,0), and extending into the first quadrant, indicating an increasing function.
(b) Use the **first derivative test** to show that \( f \) does not have a maximum or a minimum at \( x = 0 \). (Hint: you may need to look up this test. Show why it is ok to apply the test to this situation.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1b1d143-3544-4565-ae01-c2b2100e15ce%2Ffba35f9d-ee40-4a14-b972-476d091ee49b%2Fvajl02_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**4. D11: I can use derivatives to solve optimization problems.**
(a) Part of the graph of \( f(x) = x + 2 \sqrt[3]{x} \) is given below.
i. Does the graph indicate that \( x = 0 \) is a critical point? Why/Why not?
ii. Use **calculus and algebra** to find all possible \( x \) values where \( f \) could possibly change direction (change from increase to decrease or vice versa). Does your algebra indicate that \( x = 0 \) is a critical point? Why/Why not?
**Graph:**
The graph shows a curve starting from the third quadrant, smoothly passing through the origin (0,0), and extending into the first quadrant, indicating an increasing function.
(b) Use the **first derivative test** to show that \( f \) does not have a maximum or a minimum at \( x = 0 \). (Hint: you may need to look up this test. Show why it is ok to apply the test to this situation.)
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Step 1: Define the problem
VIEWStep 2: Define the critical point at x=0
VIEWStep 3: Calculate the derivative of function
VIEWStep 4: Calculate the critical values
VIEWStep 5: Calculate the sign table
VIEWStep 6: Calculate the increasing and decreasing intervals
VIEWStep 7: Calculate the maximum and minimum at x=0
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