Basic (1-7)
▶1. What is a maximum point of a function?
Answer: The highest point on the graph of a function in a given interval.
Step-by-step Explanation:
1. A maximum point is where the function reaches its greatest value in a region.
2. If you draw the graph, the maximum is the "peak" or "top" of the curve.
3. At this point, the slope (derivative) is zero, and the curve changes from increasing to decreasing.
▶2. What is a minimum point of a function?
Answer: The lowest point on the graph of a function in a given interval.
Step-by-step Explanation:
1. A minimum point is where the function reaches its smallest value in a region.
2. On the graph, the minimum is the "valley" or "bottom" of the curve.
3. At this point, the slope (derivative) is zero, and the curve changes from decreasing to increasing.
▶3. What is a critical point?
Answer: A point where the derivative is zero or undefined.
Step-by-step Explanation:
1. A critical point is where the function might have a maximum, minimum, or neither.
2. It occurs when the first derivative (slope) is zero or does not exist.
3. We check these points to find maxima and minima.
▶4. Find the critical points of f(x) = x² - 4x + 3.
Answer: x = 2
Step-by-step Explanation:
1. Find the first derivative: f'(x) = 2x - 4.
2. Set f'(x) = 0: 2x - 4 = 0.
3. Solve for x: 2x = 4 → x = 2.
4. So, the critical point is at x = 2.
▶5. Find the maximum or minimum value of f(x) = -x² + 4x - 1.
Answer: Maximum at x = 2, value = 3
Step-by-step Explanation:
1. Find the first derivative: f'(x) = -2x + 4.
2. Set f'(x) = 0: -2x + 4 = 0 → x = 2.
3. Find the second derivative: f''(x) = -2 (which is negative, so it's a maximum).
4. Substitute x = 2 into f(x): f(2) = -(2)² + 4×2 - 1 = -4 + 8 - 1 = 3.
5. So, maximum value is 3 at x = 2.
▶6. Find the minimum value of f(x) = x² + 6x + 5.
Answer: Minimum at x = -3, value = -4
Step-by-step Explanation:
1. Find the first derivative: f'(x) = 2x + 6.
2. Set f'(x) = 0: 2x + 6 = 0 → x = -3.
3. Find the second derivative: f''(x) = 2 (which is positive, so it's a minimum).
4. Substitute x = -3 into f(x): f(-3) = (-3)² + 6×(-3) + 5 = 9 - 18 + 5 = -4.
5. So, minimum value is -4 at x = -3.
▶7. Find the maximum value of f(x) = -2x² + 8x + 1.
Answer: Maximum at x = 2, value = 9
Step-by-step Explanation:
1. Find the first derivative: f'(x) = -4x + 8.
2. Set f'(x) = 0: -4x + 8 = 0 → x = 2.
3. Find the second derivative: f''(x) = -4 (which is negative, so it's a maximum).
4. Substitute x = 2 into f(x): f(2) = -2×(2)² + 8×2 + 1 = -8 + 16 + 1 = 9.
5. So, maximum value is 9 at x = 2.
Moderate (8-14)
▶8. Find the minimum value of f(x) = x² - 10x + 25.
Answer: Minimum at x = 5, value = 0
Step-by-step Explanation:
1. Find the first derivative: f'(x) = 2x - 10.
2. Set f'(x) = 0: 2x - 10 = 0 → x = 5.
3. Find the second derivative: f''(x) = 2 (which is positive, so it's a minimum).
4. Substitute x = 5 into f(x): f(5) = (5)² - 10×5 + 25 = 25 - 50 + 25 = 0.
5. So, minimum value is 0 at x = 5.
▶9. Find the maximum value of f(x) = -x² + 6x - 8.
Answer: Maximum at x = 3, value = 1
Step-by-step Explanation:
1. Find the first derivative: f'(x) = -2x + 6.
2. Set f'(x) = 0: -2x + 6 = 0 → x = 3.
3. Find the second derivative: f''(x) = -2 (which is negative, so it's a maximum).
4. Substitute x = 3 into f(x): f(3) = -(3)² + 6×3 - 8 = -9 + 18 - 8 = 1.
5. So, maximum value is 1 at x = 3.
▶10. Find the minimum value of f(x) = 2x² - 12x + 20.
Answer: Minimum at x = 3, value = 2
Step-by-step Explanation:
1. Find the first derivative: f'(x) = 4x - 12.
2. Set f'(x) = 0: 4x - 12 = 0 → x = 3.
3. Find the second derivative: f''(x) = 4 (which is positive, so it's a minimum).
4. Substitute x = 3 into f(x): f(3) = 2×(3)² - 12×3 + 20 = 18 - 36 + 20 = 2.
5. So, minimum value is 2 at x = 3.
▶11. Find the maximum value of f(x) = -3x² + 12x - 7.
Answer: Maximum at x = 2, value = 5
Step-by-step Explanation:
1. Find the first derivative: f'(x) = -6x + 12.
2. Set f'(x) = 0: -6x + 12 = 0 → x = 2.
3. Find the second derivative: f''(x) = -6 (which is negative, so it's a maximum).
4. Substitute x = 2 into f(x): f(2) = -3×(2)² + 12×2 - 7 = -12 + 24 - 7 = 5.
5. So, maximum value is 5 at x = 2.
▶12. Find the minimum value of f(x) = x² + 2x + 1.
Answer: Minimum at x = -1, value = 0
Step-by-step Explanation:
1. Find the first derivative: f'(x) = 2x + 2.
2. Set f'(x) = 0: 2x + 2 = 0 → x = -1.
3. Find the second derivative: f''(x) = 2 (which is positive, so it's a minimum).
4. Substitute x = -1 into f(x): f(-1) = (-1)² + 2×(-1) + 1 = 1 - 2 + 1 = 0.
5. So, minimum value is 0 at x = -1.
▶13. Find the maximum value of f(x) = -x² + 2x + 3.
Answer: Maximum at x = 1, value = 4
Step-by-step Explanation:
1. Find the first derivative: f'(x) = -2x + 2.
2. Set f'(x) = 0: -2x + 2 = 0 → x = 1.
3. Find the second derivative: f''(x) = -2 (which is negative, so it's a maximum).
4. Substitute x = 1 into f(x): f(1) = -1 + 2 + 3 = 4.
5. So, maximum value is 4 at x = 1.
▶14. Find the minimum value of f(x) = 3x² - 18x + 27.
Answer: Minimum at x = 3, value = 0
Step-by-step Explanation:
1. Find the first derivative: f'(x) = 6x - 18.
2. Set f'(x) = 0: 6x - 18 = 0 → x = 3.
3. Find the second derivative: f''(x) = 6 (which is positive, so it's a minimum).
4. Substitute x = 3 into f(x): f(3) = 3×(3)² - 18×3 + 27 = 27 - 54 + 27 = 0.
5. So, minimum value is 0 at x = 3.
▶15. Find the maximum value of f(x) = -2x² + 4x + 6.
Answer: Maximum at x = 1, value = 8
Step-by-step Explanation:
1. Find the first derivative: f'(x) = -4x + 4.
2. Set f'(x) = 0: -4x + 4 = 0 → x = 1.
3. Find the second derivative: f''(x) = -4 (which is negative, so it's a maximum).
4. Substitute x = 1 into f(x): f(1) = -2×(1)² + 4×1 + 6 = -2 + 4 + 6 = 8.
5. So, maximum value is 8 at x = 1.
Advanced (16-20)
▶16. Find the minimum value of f(x) = x² - 8x + 16.
Answer: Minimum at x = 4, value = 0
Step-by-step Explanation:
1. Find the first derivative: f'(x) = 2x - 8.
2. Set f'(x) = 0: 2x - 8 = 0 → x = 4.
3. Find the second derivative: f''(x) = 2 (which is positive, so it's a minimum).
4. Substitute x = 4 into f(x): f(4) = (4)² - 8×4 + 16 = 16 - 32 + 16 = 0.
5. So, minimum value is 0 at x = 4.
▶17. Find the maximum value of f(x) = -x² + 10x - 21.
Answer: Maximum at x = 5, value = 4
Step-by-step Explanation:
1. Find the first derivative: f'(x) = -2x + 10.
2. Set f'(x) = 0: -2x + 10 = 0 → x = 5.
3. Find the second derivative: f''(x) = -2 (which is negative, so it's a maximum).
4. Substitute x = 5 into f(x): f(5) = -25 + 50 - 21 = 4.
5. So, maximum value is 4 at x = 5.
▶18. Find the minimum value of f(x) = x² + 4x + 4.
Answer: Minimum at x = -2, value = 0
Step-by-step Explanation:
1. Find the first derivative: f'(x) = 2x + 4.
2. Set f'(x) = 0: 2x + 4 = 0 → x = -2.
3. Find the second derivative: f''(x) = 2 (which is positive, so it's a minimum).
4. Substitute x = -2 into f(x): f(-2) = (-2)² + 4×(-2) + 4 = 4 - 8 + 4 = 0.
5. So, minimum value is 0 at x = -2.
▶19. Find the maximum value of f(x) = -x² + 12x - 36.
Answer: Maximum at x = 6, value = 0
Step-by-step Explanation:
1. Find the first derivative: f'(x) = -2x + 12.
2. Set f'(x) = 0: -2x + 12 = 0 → x = 6.
3. Find the second derivative: f''(x) = -2 (which is negative, so it's a maximum).
4. Substitute x = 6 into f(x): f(6) = -36 + 72 - 36 = 0.
5. So, maximum value is 0 at x = 6.
▶20. Find the minimum value of f(x) = 2x² - 16x + 32.
Answer: Minimum at x = 4, value = 0
Step-by-step Explanation:
1. Find the first derivative: f'(x) = 4x - 16.
2. Set f'(x) = 0: 4x - 16 = 0 → x = 4.
3. Find the second derivative: f''(x) = 4 (which is positive, so it's a minimum).
4. Substitute x = 4 into f(x): f(4) = 2×(4)² - 16×4 + 32 = 32 - 64 + 32 = 0.
5. So, minimum value is 0 at x = 4.