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Consider the function f(x) = x^3 - x^2 - 2x + 1. Which numerical method can be used to find the maximum value of f(x) in the interval [0, 2]?
- a)Newton's method
- b)Bisection method
- c)Simpson's formula
- d)Trapezoidal rule
Correct answer is option 'D'. Can you explain this answer?
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Consider the function f(x) = x^3 - x^2 - 2x + 1. Which numerical metho...
To find the maximum value of f(x) = x^3 - x^2 - 2x + 1 in the interval [0, 2], we can use the Trapezoidal rule. By approximating the integral of the function over the interval using trapezoids, we can determine the maximum value. The Trapezoidal rule can handle both continuous and discrete functions.
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Consider the function f(x) = x^3 - x^2 - 2x + 1. Which numerical metho...
Using the Trapezoidal Rule to find the maximum value of f(x) in the interval [0, 2]
The Trapezoidal Rule is a numerical method for approximating definite integrals. In this case, we can use it to find the maximum value of the function f(x) = x^3 - x^2 - 2x in the interval [0, 2].
1. Find the critical points of f(x)
To find the maximum value of a function, we first need to find the critical points. These are the points where the derivative of the function is equal to zero or undefined.
The derivative of f(x) = x^3 - x^2 - 2x is given by:
f'(x) = 3x^2 - 2x - 2
Setting f'(x) equal to zero and solving for x, we get:
3x^2 - 2x - 2 = 0
This is a quadratic equation, which can be solved using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 3, b = -2, and c = -2. Plugging these values into the quadratic formula, we get:
x = (-(-2) ± √((-2)^2 - 4(3)(-2))) / (2(3))
= (2 ± √(4 + 24)) / 6
= (2 ± √28) / 6
= (2 ± 2√7) / 6
Simplifying further, we have:
x = (1 ± √7) / 3
So the critical points of f(x) are x = (1 + √7) / 3 and x = (1 - √7) / 3.
2. Calculate the values of f(x) at the critical points and endpoints
Next, we need to calculate the values of f(x) at the critical points and endpoints of the interval [0, 2].
f(0) = (0)^3 - (0)^2 - 2(0) = 0
f(2) = (2)^3 - (2)^2 - 2(2) = 0
f((1 + √7) / 3) ≈ 0.198
f((1 - √7) / 3) ≈ -2.198
3. Find the maximum value of f(x)
Since we have calculated the values of f(x) at the critical points and endpoints, we can compare these values to find the maximum value of f(x) in the interval [0, 2].
From the calculations, we can see that the maximum value of f(x) in the interval [0, 2] is approximately 0.198, which occurs at x ≈ (1 + √7) / 3.
Therefore, the correct answer is option 'D' - Trapezoidal rule.
The Trapezoidal Rule is a numerical method for approximating definite integrals. In this case, we can use it to find the maximum value of the function f(x) = x^3 - x^2 - 2x in the interval [0, 2].
1. Find the critical points of f(x)
To find the maximum value of a function, we first need to find the critical points. These are the points where the derivative of the function is equal to zero or undefined.
The derivative of f(x) = x^3 - x^2 - 2x is given by:
f'(x) = 3x^2 - 2x - 2
Setting f'(x) equal to zero and solving for x, we get:
3x^2 - 2x - 2 = 0
This is a quadratic equation, which can be solved using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 3, b = -2, and c = -2. Plugging these values into the quadratic formula, we get:
x = (-(-2) ± √((-2)^2 - 4(3)(-2))) / (2(3))
= (2 ± √(4 + 24)) / 6
= (2 ± √28) / 6
= (2 ± 2√7) / 6
Simplifying further, we have:
x = (1 ± √7) / 3
So the critical points of f(x) are x = (1 + √7) / 3 and x = (1 - √7) / 3.
2. Calculate the values of f(x) at the critical points and endpoints
Next, we need to calculate the values of f(x) at the critical points and endpoints of the interval [0, 2].
f(0) = (0)^3 - (0)^2 - 2(0) = 0
f(2) = (2)^3 - (2)^2 - 2(2) = 0
f((1 + √7) / 3) ≈ 0.198
f((1 - √7) / 3) ≈ -2.198
3. Find the maximum value of f(x)
Since we have calculated the values of f(x) at the critical points and endpoints, we can compare these values to find the maximum value of f(x) in the interval [0, 2].
From the calculations, we can see that the maximum value of f(x) in the interval [0, 2] is approximately 0.198, which occurs at x ≈ (1 + √7) / 3.
Therefore, the correct answer is option 'D' - Trapezoidal rule.
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Consider the function f(x) = x^3 - x^2 - 2x + 1. Which numerical method can be used to find the maximum value of f(x) in the interval [0, 2]?a)Newtons methodb)Bisection methodc)Simpsons formulad)Trapezoidal ruleCorrect answer is option 'D'. Can you explain this answer?
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