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The value of ‘a’ for which x3 - 3x + a = 0 has two distinct roots in [0, 1] is given by
- a)-1
- b)1
- c)3
- d)does not exists
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
The value of ‘a’ for which x3 - 3x + a = 0 has two distinc...
Let α, β ∈[0,1].f (x) is continuous on [a,b] & differentiable on (a,b) and f (α) = f (β) = 0
∴ c ∈ (α, β) such that f' (c) = 0 ⇒ c = ±1∉ (0,1)
∴ c ∈ (α, β) such that f' (c) = 0 ⇒ c = ±1∉ (0,1)
Most Upvoted Answer
The value of ‘a’ for which x3 - 3x + a = 0 has two distinc...
Question Analysis:
We are given a cubic equation x^3 - 3x + a = 0 and we need to find the value of a for which the equation has two distinct roots in the interval [0, 1].
Solution:
To find the value of a, we can use the concept of Rolle's theorem. According to Rolle's theorem, for a function f(x) to have two distinct roots in the interval [0, 1], the function must have a local maximum or minimum between the two roots.
Step 1: Find the derivative of the given cubic equation.
The derivative of x^3 - 3x + a with respect to x is:
f'(x) = 3x^2 - 3
Step 2: Find the critical points of the function.
To find the critical points, we need to solve the equation f'(x) = 0.
3x^2 - 3 = 0
3x^2 = 3
x^2 = 1
x = ±1
Step 3: Check if the critical points lie in the interval [0, 1].
Since the interval is [0, 1], we only need to check if x = 1 is a critical point.
f'(1) = 3(1)^2 - 3 = 3 - 3 = 0
Step 4: Apply Rolle's theorem.
Since the derivative has a critical point at x = 1, there must be a local maximum or minimum between the two roots of the cubic equation.
Step 5: Conclusion.
From the above analysis, we can conclude that there exists a value of a for which the equation x^3 - 3x + a = 0 has two distinct roots in the interval [0, 1]. Therefore, the correct answer is option D - "Does not exist".
We are given a cubic equation x^3 - 3x + a = 0 and we need to find the value of a for which the equation has two distinct roots in the interval [0, 1].
Solution:
To find the value of a, we can use the concept of Rolle's theorem. According to Rolle's theorem, for a function f(x) to have two distinct roots in the interval [0, 1], the function must have a local maximum or minimum between the two roots.
Step 1: Find the derivative of the given cubic equation.
The derivative of x^3 - 3x + a with respect to x is:
f'(x) = 3x^2 - 3
Step 2: Find the critical points of the function.
To find the critical points, we need to solve the equation f'(x) = 0.
3x^2 - 3 = 0
3x^2 = 3
x^2 = 1
x = ±1
Step 3: Check if the critical points lie in the interval [0, 1].
Since the interval is [0, 1], we only need to check if x = 1 is a critical point.
f'(1) = 3(1)^2 - 3 = 3 - 3 = 0
Step 4: Apply Rolle's theorem.
Since the derivative has a critical point at x = 1, there must be a local maximum or minimum between the two roots of the cubic equation.
Step 5: Conclusion.
From the above analysis, we can conclude that there exists a value of a for which the equation x^3 - 3x + a = 0 has two distinct roots in the interval [0, 1]. Therefore, the correct answer is option D - "Does not exist".
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