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Consider function f(x) = (x2- 4)2 where x is a real number. Then the function has
- a)Only one minimum
- b)Only two minima
- c)Three minima
- d)Three maxima
Correct answer is option 'B'. Can you explain this answer?
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Consider function f(x) = (x2- 4)2 where x is a real number. Then the f...
∴ There is only one maxima and only two minima for this function.
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Consider function f(x) = (x2- 4)2 where x is a real number. Then the f...
Explanation:
Function Analysis:
- The function f(x) = (x^2 - 4)^2 can be simplified to f(x) = (x^4 - 8x^2 + 16).
- To find the minimum points of the function, we need to find where the derivative of the function is equal to zero.
Finding the Derivative:
- f'(x) = d/dx (x^4 - 8x^2 + 16)
- f'(x) = 4x^3 - 16x
- Setting f'(x) = 0 to find critical points:
4x^3 - 16x = 0
4x(x^2 - 4) = 0
x(x + 2)(x - 2) = 0
- This gives critical points at x = -2, x = 0, and x = 2.
Classifying the Critical Points:
- To determine whether these critical points are minima or maxima, we can use the second derivative test.
- f''(x) = d^2/dx^2 (4x^3 - 16x)
- f''(x) = 12x^2 - 16
- For x = -2, f''(-2) = 56 > 0, so it is a local minimum.
- For x = 0, f''(0) = -16 < 0,="" so="" it="" is="" a="" local="" />
- For x = 2, f''(2) = 56 > 0, so it is a local minimum.
Conclusion:
- The function f(x) = (x^2 - 4)^2 has two minima at x = -2 and x = 2.
Function Analysis:
- The function f(x) = (x^2 - 4)^2 can be simplified to f(x) = (x^4 - 8x^2 + 16).
- To find the minimum points of the function, we need to find where the derivative of the function is equal to zero.
Finding the Derivative:
- f'(x) = d/dx (x^4 - 8x^2 + 16)
- f'(x) = 4x^3 - 16x
- Setting f'(x) = 0 to find critical points:
4x^3 - 16x = 0
4x(x^2 - 4) = 0
x(x + 2)(x - 2) = 0
- This gives critical points at x = -2, x = 0, and x = 2.
Classifying the Critical Points:
- To determine whether these critical points are minima or maxima, we can use the second derivative test.
- f''(x) = d^2/dx^2 (4x^3 - 16x)
- f''(x) = 12x^2 - 16
- For x = -2, f''(-2) = 56 > 0, so it is a local minimum.
- For x = 0, f''(0) = -16 < 0,="" so="" it="" is="" a="" local="" />
- For x = 2, f''(2) = 56 > 0, so it is a local minimum.
Conclusion:
- The function f(x) = (x^2 - 4)^2 has two minima at x = -2 and x = 2.
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Consider function f(x) = (x2- 4)2 where x is a real number. Then the function hasa)Only one minimumb)Only two minimac)Three minimad)Three maximaCorrect answer is option 'B'. Can you explain this answer?
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Consider function f(x) = (x2- 4)2 where x is a real number. Then the function hasa)Only one minimumb)Only two minimac)Three minimad)Three maximaCorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider function f(x) = (x2- 4)2 where x is a real number. Then the function hasa)Only one minimumb)Only two minimac)Three minimad)Three maximaCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider function f(x) = (x2- 4)2 where x is a real number. Then the function hasa)Only one minimumb)Only two minimac)Three minimad)Three maximaCorrect answer is option 'B'. Can you explain this answer?.
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