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Which of the following functions will have a minimum value at x = –3?
- a)f(x) = 2x3 – 4x + 3
- b)f(x) = 4x4 – 3x + 5
- c)f(x) = x6 – 2x – 6
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Which of the following functions will have a minimum value at x = R...
If you differentiate each function with respect to x, and equate it to 0 you would see that for none
of the three options will get you a value of x = –3 as its solution. Thus, option (d) viz. None of
these is correct.
of the three options will get you a value of x = –3 as its solution. Thus, option (d) viz. None of
these is correct.
Most Upvoted Answer
Which of the following functions will have a minimum value at x = R...
Explanation:
To determine which function will have a minimum value at x = 3, we need to analyze the behavior of each function in the vicinity of x = 3.
Function a: f(x) = 2x^3 - 4x + 3
- To find the minimum, we need to take the derivative of the function and set it equal to zero.
- Taking the derivative of f(x) with respect to x, we get: f'(x) = 6x^2 - 4
- Setting f'(x) = 0 and solving for x, we get: 6x^2 - 4 = 0
- Solving this quadratic equation, we find: x = ±√(4/6) = ±√(2/3)
- Since the roots of the derivative are not equal to 3, the function does not have a minimum at x = 3.
Function b: f(x) = 4x^4 - 3x + 5
- Taking the derivative of f(x) with respect to x, we get: f'(x) = 16x^3 - 3
- Setting f'(x) = 0 and solving for x, we get: 16x^3 - 3 = 0
- We can use numerical methods or approximations to find the root, but it is evident that the root is not 3.
- Therefore, the function does not have a minimum at x = 3.
Function c: f(x) = x^6 - 2x + 6
- Taking the derivative of f(x) with respect to x, we get: f'(x) = 6x^5 - 2
- Setting f'(x) = 0 and solving for x, we get: 6x^5 - 2 = 0
- We can use numerical methods or approximations to find the root, but it is evident that the root is not 3.
- Therefore, the function does not have a minimum at x = 3.
Conclusion:
None of the given functions have a minimum at x = 3.
To determine which function will have a minimum value at x = 3, we need to analyze the behavior of each function in the vicinity of x = 3.
Function a: f(x) = 2x^3 - 4x + 3
- To find the minimum, we need to take the derivative of the function and set it equal to zero.
- Taking the derivative of f(x) with respect to x, we get: f'(x) = 6x^2 - 4
- Setting f'(x) = 0 and solving for x, we get: 6x^2 - 4 = 0
- Solving this quadratic equation, we find: x = ±√(4/6) = ±√(2/3)
- Since the roots of the derivative are not equal to 3, the function does not have a minimum at x = 3.
Function b: f(x) = 4x^4 - 3x + 5
- Taking the derivative of f(x) with respect to x, we get: f'(x) = 16x^3 - 3
- Setting f'(x) = 0 and solving for x, we get: 16x^3 - 3 = 0
- We can use numerical methods or approximations to find the root, but it is evident that the root is not 3.
- Therefore, the function does not have a minimum at x = 3.
Function c: f(x) = x^6 - 2x + 6
- Taking the derivative of f(x) with respect to x, we get: f'(x) = 6x^5 - 2
- Setting f'(x) = 0 and solving for x, we get: 6x^5 - 2 = 0
- We can use numerical methods or approximations to find the root, but it is evident that the root is not 3.
- Therefore, the function does not have a minimum at x = 3.
Conclusion:
None of the given functions have a minimum at x = 3.
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