Where f′ is zero or undefined: the candidates for maxima and minima. 70 problems with worked solutions; in 70 of them every
equation is proved by a computer algebra system.
20 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
\[ 3 x^{2} - 3 x - 18 = \left(x + 2\right) \left(3 x - 9\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = -2 and x = 3.
Reviewed
Answer \( x = -2, 3 \)
Top: the graph of f = x³ - 3x²/2 - 18x - 2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3x² - 3x - 18. The tangent is flat at x = -2, 3.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
\[ 6 x^{2} + 6 x - 72 = \left(x + 4\right) \left(6 x - 18\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = -4 and x = 3.
Reviewed
Answer \( x = -4, 3 \)
Top: the graph of f = 2x³ + 3x² - 72x + 5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 6x² + 6x - 72. The tangent is flat at x = -4, 3.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -2x³ - 9x² + 60x - 4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -6x² - 18x + 60. The tangent is flat at x = -5, 2.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -x³ - 6x² + 15x + 2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3x² - 12x + 15. The tangent is flat at x = -5, 1.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
\[ 6 x^{2} + 12 x - 18 = \left(x + 3\right) \left(6 x - 6\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = -3 and x = 1.
Reviewed
Answer \( x = -3, 1 \)
Top: the graph of f = 2x³ + 6x² - 18x - 5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 6x² + 12x - 18. The tangent is flat at x = -3, 1.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
\[ 3 x^{2} - 6 x - 24 = \left(x + 2\right) \left(3 x - 12\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = -2 and x = 4.
Reviewed
Answer \( x = -2, 4 \)
Top: the graph of f = x³ - 3x² - 24x + 2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3x² - 6x - 24. The tangent is flat at x = -2, 4.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -x³ + 6x² + 1. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3x² + 12x. The tangent is flat at x = 0, 4.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -x³ + 6x² + 15x - 2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3x² + 12x + 15. The tangent is flat at x = -1, 5.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = x³ - 3x + 5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3x² - 3. The tangent is flat at x = -1, 1.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
\[ - 6 x^{2} - 54 x - 120 = \left(- 6 x - 24\right) \left(x + 5\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = -5 and x = -4.
Reviewed
Answer \( x = -5, -4 \)
Top: the graph of f = -2x³ - 27x² - 120x + 5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -6x² - 54x - 120. The tangent is flat at x = -5, -4.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
\[ 9 x^{2} + 81 x + 180 = \left(x + 5\right) \left(9 x + 36\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = -5 and x = -4.
Reviewed
Answer \( x = -5, -4 \)
Top: the graph of f = 3x³ + 81x²/2 + 180x + 4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 9x² + 81x + 180. The tangent is flat at x = -5, -4.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -2x³ + 3x² + 2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -6x² + 6x. The tangent is flat at x = 0, 1.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
\[ 6 x^{2} - 48 x + 90 = \left(x - 3\right) \left(6 x - 30\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = 3 and x = 5.
Reviewed
Answer \( x = 3, 5 \)
Top: the graph of f = 2x³ - 24x² + 90x - 1. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 6x² - 48x + 90. The tangent is flat at x = 3, 5.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -2x³ + 24x - 5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 24 - 6x². The tangent is flat at x = -2, 2.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -3x³ + 9x²/2 + 180x - 1. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -9x² + 9x + 180. The tangent is flat at x = -4, 5.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -2x³ - 3x² + 12x + 3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -6x² - 6x + 12. The tangent is flat at x = -2, 1.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -3x³ + 81x + 3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 81 - 9x². The tangent is flat at x = -3, 3.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
\[ 6 x^{2} + 18 x + 12 = \left(x + 2\right) \left(6 x + 6\right) \]
Factor f'(x).✓ Proved
f'(x) = 0 at x = -2 and x = -1.
Reviewed
Answer \( x = -2, -1 \)
Top: the graph of f = 2x³ + 9x² + 12x - 5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 6x² + 18x + 12. The tangent is flat at x = -2, -1.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = -3x³ + 27x² - 45x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -9x² + 54x - 45. The tangent is flat at x = 1, 5.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Top: the graph of f = x³ - 27x - 4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3x² - 27. The tangent is flat at x = -3, 3.
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify