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The function f(x) = x3 - 6x2 + 9x + 25 has
- a)a maxima at x= 1 and a minima at x = 3
- b)a maxima at x = 3 and a minima at x = 1
- c)no maxima, but a minima at x = 1
- d)a maxima at x = 1, but no minima
Correct answer is option 'A'. Can you explain this answer?
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The function f(x) = x3 - 6x2 + 9x + 25 hasa)a maxima at x= 1 and a min...
Given function: f(x) = x^3 - 6x^2 + 9x + 25
To find the maxima and minima of this function, we need to differentiate it with respect to x and equate it to zero.
f'(x) = 3x^2 - 12x + 9
Setting f'(x) = 0, we get:
3x^2 - 12x + 9 = 0
Dividing by 3, we get:
x^2 - 4x + 3 = 0
Factorizing, we get:
(x - 1)(x - 3) = 0
Therefore, x = 1 or x = 3
To determine whether these points correspond to maxima or minima, we need to find the second derivative of the function.
f''(x) = 6x - 12
Substituting x = 1, we get:
f''(1) = 6(1) - 12 = -6
Since f''(1) is negative, the point x = 1 corresponds to a maxima.
Substituting x = 3, we get:
f''(3) = 6(3) - 12 = 6
Since f''(3) is positive, the point x = 3 corresponds to a minima.
Therefore, the function f(x) = x^3 - 6x^2 + 9x + 25 has a maxima at x = 1 and a minima at x = 3. The correct answer is option A.
To find the maxima and minima of this function, we need to differentiate it with respect to x and equate it to zero.
f'(x) = 3x^2 - 12x + 9
Setting f'(x) = 0, we get:
3x^2 - 12x + 9 = 0
Dividing by 3, we get:
x^2 - 4x + 3 = 0
Factorizing, we get:
(x - 1)(x - 3) = 0
Therefore, x = 1 or x = 3
To determine whether these points correspond to maxima or minima, we need to find the second derivative of the function.
f''(x) = 6x - 12
Substituting x = 1, we get:
f''(1) = 6(1) - 12 = -6
Since f''(1) is negative, the point x = 1 corresponds to a maxima.
Substituting x = 3, we get:
f''(3) = 6(3) - 12 = 6
Since f''(3) is positive, the point x = 3 corresponds to a minima.
Therefore, the function f(x) = x^3 - 6x^2 + 9x + 25 has a maxima at x = 1 and a minima at x = 3. The correct answer is option A.
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The function f(x) = x3 - 6x2 + 9x + 25 hasa)a maxima at x= 1 and a minima at x = 3b)a maxima at x = 3 and a minima at x = 1c)no maxima, but a minima at x = 1d)a maxima at x = 1, but no minimaCorrect answer is option 'A'. Can you explain this answer?
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