Let f(x) = x³ + 6x² (a) Find f'(x) = (b) Find f''(x) = 15x + 4. (c) On what set is f increasing? interval of increase: (d) On what set is f decreasing? interval of decrease: (e) On what set is f concave downward? interval of downward concavity: (f) On what set is f concave upward? interval of upward concavity:
Let f(x) = x³ + 6x² (a) Find f'(x) = (b) Find f''(x) = 15x + 4. (c) On what set is f increasing? interval of increase: (d) On what set is f decreasing? interval of decrease: (e) On what set is f concave downward? interval of downward concavity: (f) On what set is f concave upward? interval of upward concavity:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Function Analysis Exercise**
Given the function \( f(x) = x^3 + 6x^2 - 15x + 4 \).
(a) Find \( f'(x) \).
\[ f'(x) = \]
(b) Find \( f''(x) \).
\[ f''(x) = \]
(c) On what interval is \( f \) increasing?
Interval of increase: \(\quad\)
(d) On what interval is \( f \) decreasing?
Interval of decrease: \(\quad\)
(e) On what interval is \( f \) concave downward?
Interval of downward concavity: \(\quad\)
(f) On what interval is \( f \) concave upward?
Interval of upward concavity: \(\quad\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4df0d9d6-a5af-48e8-af07-cbf992b42454%2F14c29a05-bdb2-4da2-b5eb-d51b15060928%2Fowd9nlc_processed.png&w=3840&q=75)
Transcribed Image Text:**Function Analysis Exercise**
Given the function \( f(x) = x^3 + 6x^2 - 15x + 4 \).
(a) Find \( f'(x) \).
\[ f'(x) = \]
(b) Find \( f''(x) \).
\[ f''(x) = \]
(c) On what interval is \( f \) increasing?
Interval of increase: \(\quad\)
(d) On what interval is \( f \) decreasing?
Interval of decrease: \(\quad\)
(e) On what interval is \( f \) concave downward?
Interval of downward concavity: \(\quad\)
(f) On what interval is \( f \) concave upward?
Interval of upward concavity: \(\quad\)
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