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If the function f (x) = 2x3 - 9ax2 + 12a2x + 1, where a > 0 , attains its maximum and minimum at p and q respectively such that p² = q, then ‘a’ equals
- a)3
- b)1
- c)2
- d)1/2
Correct answer is option 'C'. Can you explain this answer?
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If the function f (x) = 2x3 - 9ax2 + 12a2x + 1, where a > 0 , atta...
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If the function f (x) = 2x3 - 9ax2 + 12a2x + 1, where a > 0 , atta...
To find the maximum and minimum of the function f(x), we need to find the critical points of the function.
Finding the derivative of f(x) with respect to x, we get:
f'(x) = 6x^2 - 18ax + 12a^2
To find the critical points, we set f'(x) equal to zero:
6x^2 - 18ax + 12a^2 = 0
Dividing both sides of the equation by 6, we get:
x^2 - 3ax + 2a^2 = 0
This is a quadratic equation in x. To solve for x, we can use the quadratic formula:
x = (-(-3a) ± √((-3a)^2 - 4(1)(2a^2))) / (2(1))
Simplifying this expression, we get:
x = (3a ± √(9a^2 - 8a^2)) / 2
x = (3a ± √(a^2)) / 2
x = (3a ± a) / 2
We have two solutions for x:
x1 = (3a + a) / 2 = 2a
x2 = (3a - a) / 2 = a
So we have two critical points: x1 = 2a and x2 = a.
To find the maximum and minimum of f(x), we substitute these critical points back into the original function:
f(2a) = 2(2a)^3 - 9a(2a)^2 + 12a^2(2a) - 1
f(2a) = 16a^3 - 36a^3 + 48a^3 - 1
f(2a) = 28a^3 - 1
f(a) = 2(a)^3 - 9a(a)^2 + 12a^2(a) - 1
f(a) = 2a^3 - 9a^3 + 12a^3 - 1
f(a) = 5a^3 - 1
Since p = q, the maximum and minimum occur at the same value of x, which means f(2a) = f(a).
Setting these two expressions equal to each other, we get:
28a^3 - 1 = 5a^3 - 1
Subtracting 5a^3 from both sides of the equation, we get:
23a^3 = 0
Dividing both sides by 23, we get:
a^3 = 0
Taking the cube root of both sides, we get:
a = 0
Therefore, the value of a that satisfies the condition is 0.
Finding the derivative of f(x) with respect to x, we get:
f'(x) = 6x^2 - 18ax + 12a^2
To find the critical points, we set f'(x) equal to zero:
6x^2 - 18ax + 12a^2 = 0
Dividing both sides of the equation by 6, we get:
x^2 - 3ax + 2a^2 = 0
This is a quadratic equation in x. To solve for x, we can use the quadratic formula:
x = (-(-3a) ± √((-3a)^2 - 4(1)(2a^2))) / (2(1))
Simplifying this expression, we get:
x = (3a ± √(9a^2 - 8a^2)) / 2
x = (3a ± √(a^2)) / 2
x = (3a ± a) / 2
We have two solutions for x:
x1 = (3a + a) / 2 = 2a
x2 = (3a - a) / 2 = a
So we have two critical points: x1 = 2a and x2 = a.
To find the maximum and minimum of f(x), we substitute these critical points back into the original function:
f(2a) = 2(2a)^3 - 9a(2a)^2 + 12a^2(2a) - 1
f(2a) = 16a^3 - 36a^3 + 48a^3 - 1
f(2a) = 28a^3 - 1
f(a) = 2(a)^3 - 9a(a)^2 + 12a^2(a) - 1
f(a) = 2a^3 - 9a^3 + 12a^3 - 1
f(a) = 5a^3 - 1
Since p = q, the maximum and minimum occur at the same value of x, which means f(2a) = f(a).
Setting these two expressions equal to each other, we get:
28a^3 - 1 = 5a^3 - 1
Subtracting 5a^3 from both sides of the equation, we get:
23a^3 = 0
Dividing both sides by 23, we get:
a^3 = 0
Taking the cube root of both sides, we get:
a = 0
Therefore, the value of a that satisfies the condition is 0.
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If the function f (x) = 2x3 - 9ax2 + 12a2x + 1, where a > 0 , attains its maximum and minimum at p and q respectively such that p² = q, then ‘a’ equalsa)3b)1c)2d)1/2Correct answer is option 'C'. Can you explain this answer?
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