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Find the critical points and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to each critical point. Let f(x) = x² + 12 x³ + =27 x² - 108x There are three critical points. If we call them C1, C2, and C31 with C1 C2 < C3, then C₁ = -4 C2 = -3 and C3 = 3 Isf a maximum or minumum at the critical points? At C1, f is Local Min At C21 f is Local Max Local Min At C3, f is These three critical give us four intervals. The left-most interval is -∞,-4, and on this interval f is Decreasing while f' is Negative The next interval (going left to right) is -4,-3. On this interval f is Increasing ✓ while f' is Next is the interval -3, 3. On this interval f is Decreasing while f' is Negative Finally, the right-most interval is 3,∞ On this interval f is Increasing ✓ while f' is Positive < Positive

Find the critical points and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to each critical point. Let f(x) = x² + 12 x³ + =27 x² - 108x There are three critical points. If we call them C1, C2, and C31 with C1 C2 < C3, then C₁ = -4 C2 = -3 and C3 = 3 Isf a maximum or minumum at the critical points? At C1, f is Local Min At C21 f is Local Max Local Min At C3, f is These three critical give us four intervals. The left-most interval is -∞,-4, and on this interval f is Decreasing while f' is Negative The next interval (going left to right) is -4,-3. On this interval f is Increasing ✓ while f' is Next is the interval -3, 3. On this interval f is Decreasing while f' is Negative Finally, the right-most interval is 3,∞ On this interval f is Increasing ✓ while f' is Positive < Positive

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.4: Definition Of The Derivative
Problem 50E
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Find the critical points and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to each critical point. Let f(x) = x² + 12 x³ + =27 x² - 108x There are three critical points. If we call them C1, C2, and C31 with C1 C2 < C3, then C₁ = -4 C2 = -3 and C3 = 3 Isf a maximum or minumum at the critical points? At C1, f is Local Min At C21 f is Local Max Local Min At C3, f is These three critical give us four intervals. The left-most interval is -∞,-4, and on this interval f is Decreasing while f' is Negative The next interval (going left to right) is -4,-3. On this interval f is Increasing ✓ while f' is Next is the interval -3, 3. On this interval f is Decreasing while f' is Negative Finally, the right-most interval is 3,∞ On this interval f is Increasing ✓ while f' is Positive < Positive
Transcribed Image Text:Find the critical points and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to each critical point. Let f(x) = x² + 12 x³ + =27 x² - 108x There are three critical points. If we call them C1, C2, and C31 with C1 C2 < C3, then C₁ = -4 C2 = -3 and C3 = 3 Isf a maximum or minumum at the critical points? At C1, f is Local Min At C21 f is Local Max Local Min At C3, f is These three critical give us four intervals. The left-most interval is -∞,-4, and on this interval f is Decreasing while f' is Negative The next interval (going left to right) is -4,-3. On this interval f is Increasing ✓ while f' is Next is the interval -3, 3. On this interval f is Decreasing while f' is Negative Finally, the right-most interval is 3,∞ On this interval f is Increasing ✓ while f' is Positive < Positive
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