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Find the maximum and minimum values of f (x) = 2x3 – 24x + 107 in the interval [1, 3].
- a)89, 69
- b)89, 75
- c)59, 56
- d)89, -9
Correct answer is option 'B'. Can you explain this answer?
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Find the maximum and minimum values of f (x) = 2x3– 24x + 107 in...
Toolbox:d/dx(x^n) = nx^n−1Maxima & Minima = f′
(x) = 0Step 1:f(x) = 2x^3−24x+107Differentiating with
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Find the maximum and minimum values of f (x) = 2x3– 24x + 107 in...
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Find the maximum and minimum values of f (x) = 2x3– 24x + 107 in...
Finding Maximum and Minimum Values of f(x) in the Interval [1, 3]
To find the maximum and minimum values of the function f(x) = 2x^3 - 24x + 107 in the interval [1, 3], we need to follow these steps:
Step 1: Find Critical Points
1. Calculate the derivative of f(x) to find critical points.
2. f'(x) = 6x^2 - 24
3. Set f'(x) = 0 to find critical points.
4. 6x^2 - 24 = 0
5. x^2 - 4 = 0
6. (x - 2)(x + 2) = 0
7. Critical points are x = 2 and x = -2.
Step 2: Evaluate Endpoints and Critical Points
1. Evaluate f(x) at x = 1, 2, 3, and the critical points x = 2, -2.
2. f(1) = 2(1)^3 - 24(1) + 107 = 85
3. f(2) = 2(2)^3 - 24(2) + 107 = 75
4. f(3) = 2(3)^3 - 24(3) + 107 = 89
5. f(-2) = 2(-2)^3 - 24(-2) + 107 = -9
Step 3: Determine Maximum and Minimum Values
1. The maximum value is 89 which occurs at x = 3.
2. The minimum value is -9 which occurs at x = -2.
Therefore, the maximum value of f(x) in the interval [1, 3] is 89 and the minimum value is -9. The correct answer is option B: 89, -9.
To find the maximum and minimum values of the function f(x) = 2x^3 - 24x + 107 in the interval [1, 3], we need to follow these steps:
Step 1: Find Critical Points
1. Calculate the derivative of f(x) to find critical points.
2. f'(x) = 6x^2 - 24
3. Set f'(x) = 0 to find critical points.
4. 6x^2 - 24 = 0
5. x^2 - 4 = 0
6. (x - 2)(x + 2) = 0
7. Critical points are x = 2 and x = -2.
Step 2: Evaluate Endpoints and Critical Points
1. Evaluate f(x) at x = 1, 2, 3, and the critical points x = 2, -2.
2. f(1) = 2(1)^3 - 24(1) + 107 = 85
3. f(2) = 2(2)^3 - 24(2) + 107 = 75
4. f(3) = 2(3)^3 - 24(3) + 107 = 89
5. f(-2) = 2(-2)^3 - 24(-2) + 107 = -9
Step 3: Determine Maximum and Minimum Values
1. The maximum value is 89 which occurs at x = 3.
2. The minimum value is -9 which occurs at x = -2.
Therefore, the maximum value of f(x) in the interval [1, 3] is 89 and the minimum value is -9. The correct answer is option B: 89, -9.
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Find the maximum and minimum values of f (x) = 2x3– 24x + 107 in the interval [1, 3].a)89, 69b)89, 75c)59, 56d)89, -9Correct answer is option 'B'. Can you explain this answer?
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