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Let a ∈ R and let f : R → R be given by f (x) = x5 – 5x + a. Then
- a)f(x) has three real roots if a > 4
- b)f(x) has only real root if a > 4
- c)f x) has three real roots if a < – 4
- d)f(x) has three real roots if – 4 < a < 4
Correct answer is option 'B,D'. Can you explain this answer?
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Verified Answer
Let a ∈ R and let f : R → R be given by f (x) = x5 – 5...
f (x) = x5 - 5 x+a
f (x) = 0 ⇒ x5 - 5 x +a = 0
⇒ a = 5x – x5 = g(x)
⇒ g(x) = 0 when x = 0, 51/4 - 51/4 and g' (x ) = 0 ⇒ x = 1, – 1
Also g (– 1) = – 4 and g(1) = 4
∴ graph of g(x) will be as shown below.
From graph
f (x) = 0 ⇒ x5 - 5 x +a = 0
⇒ a = 5x – x5 = g(x)
⇒ g(x) = 0 when x = 0, 51/4 - 51/4 and g' (x ) = 0 ⇒ x = 1, – 1
Also g (– 1) = – 4 and g(1) = 4
∴ graph of g(x) will be as shown below.
From graph
if a ∈ ( -4,-4)
then g(x) = a or f (x) = 0 has 3 real roots If a > 4 or a < – 4
then f(x) = 0 has only one real root.
∴ (b) and (d) are the correct options.
then g(x) = a or f (x) = 0 has 3 real roots If a > 4 or a < – 4
then f(x) = 0 has only one real root.
∴ (b) and (d) are the correct options.
Most Upvoted Answer
Let a ∈ R and let f : R → R be given by f (x) = x5 – 5...
Understanding the Function
The function given is \( f(x) = x^5 - 5x + a \). To analyze the roots, we need to consider the behavior of the function based on its derivative.
Derivative and Critical Points
The derivative of the function is:
\[
f'(x) = 5x^4 - 5 = 5(x^4 - 1) = 5(x^2 - 1)(x^2 + 1)
\]
This gives us critical points at \( x = -1, 1 \). The function changes monotonically between these points.
Behavior at Critical Points
- **At \( x = -1 \)**:
\[
f(-1) = (-1)^5 - 5(-1) + a = -1 + 5 + a = 4 + a
\]
- **At \( x = 1 \)**:
\[
f(1) = 1^5 - 5(1) + a = 1 - 5 + a = -4 + a
\]
Conditions for Roots
To find the conditions for the number of real roots:
- **Three Real Roots**:
For \( f(x) \) to have three real roots, \( f(-1) \) and \( f(1) \) must be of opposite signs:
- \( (4 + a) > 0 \) and \( (-4 + a) < 0="" />
- This leads to:
- \( a > -4 \)
- \( a < 4="" />
- Thus, \( -4 < a="" />< 4="" \)="" ensures="" three="" real="" />
- **Only One Real Root**:
For the function to have only one real root, the function must be either completely above or below the x-axis. Thus:
- If \( a > 4 \), then \( f(-1) > 0 \) and \( f(1) > 0 \) indicating a single real root.
Conclusion
The correct options based on the derived conditions are:
- **(B)**: \( f(x) \) has only one real root if \( a > 4 \).
- **(D)**: \( f(x) \) has three real roots if \( -4 < a="" />< 4="" \).="" 4="" />
The function given is \( f(x) = x^5 - 5x + a \). To analyze the roots, we need to consider the behavior of the function based on its derivative.
Derivative and Critical Points
The derivative of the function is:
\[
f'(x) = 5x^4 - 5 = 5(x^4 - 1) = 5(x^2 - 1)(x^2 + 1)
\]
This gives us critical points at \( x = -1, 1 \). The function changes monotonically between these points.
Behavior at Critical Points
- **At \( x = -1 \)**:
\[
f(-1) = (-1)^5 - 5(-1) + a = -1 + 5 + a = 4 + a
\]
- **At \( x = 1 \)**:
\[
f(1) = 1^5 - 5(1) + a = 1 - 5 + a = -4 + a
\]
Conditions for Roots
To find the conditions for the number of real roots:
- **Three Real Roots**:
For \( f(x) \) to have three real roots, \( f(-1) \) and \( f(1) \) must be of opposite signs:
- \( (4 + a) > 0 \) and \( (-4 + a) < 0="" />
- This leads to:
- \( a > -4 \)
- \( a < 4="" />
- Thus, \( -4 < a="" />< 4="" \)="" ensures="" three="" real="" />
- **Only One Real Root**:
For the function to have only one real root, the function must be either completely above or below the x-axis. Thus:
- If \( a > 4 \), then \( f(-1) > 0 \) and \( f(1) > 0 \) indicating a single real root.
Conclusion
The correct options based on the derived conditions are:
- **(B)**: \( f(x) \) has only one real root if \( a > 4 \).
- **(D)**: \( f(x) \) has three real roots if \( -4 < a="" />< 4="" \).="" 4="" />
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Let a ∈ R and let f : R → R be given by f (x) = x5 – 5x + a. Thena)f(x) has three real roots if a > 4b)f(x) has only real root if a > 4c)f x) has three real roots if a < – 4d)f(x) has three real roots if – 4 < a < 4Correct answer is option 'B,D'. Can you explain this answer?
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Let a ∈ R and let f : R → R be given by f (x) = x5 – 5x + a. Thena)f(x) has three real roots if a > 4b)f(x) has only real root if a > 4c)f x) has three real roots if a < – 4d)f(x) has three real roots if – 4 < a < 4Correct answer is option 'B,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let a ∈ R and let f : R → R be given by f (x) = x5 – 5x + a. Thena)f(x) has three real roots if a > 4b)f(x) has only real root if a > 4c)f x) has three real roots if a < – 4d)f(x) has three real roots if – 4 < a < 4Correct answer is option 'B,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a ∈ R and let f : R → R be given by f (x) = x5 – 5x + a. Thena)f(x) has three real roots if a > 4b)f(x) has only real root if a > 4c)f x) has three real roots if a < – 4d)f(x) has three real roots if – 4 < a < 4Correct answer is option 'B,D'. Can you explain this answer?.
Let a ∈ R and let f : R → R be given by f (x) = x5 – 5x + a. Thena)f(x) has three real roots if a > 4b)f(x) has only real root if a > 4c)f x) has three real roots if a < – 4d)f(x) has three real roots if – 4 < a < 4Correct answer is option 'B,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let a ∈ R and let f : R → R be given by f (x) = x5 – 5x + a. Thena)f(x) has three real roots if a > 4b)f(x) has only real root if a > 4c)f x) has three real roots if a < – 4d)f(x) has three real roots if – 4 < a < 4Correct answer is option 'B,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a ∈ R and let f : R → R be given by f (x) = x5 – 5x + a. Thena)f(x) has three real roots if a > 4b)f(x) has only real root if a > 4c)f x) has three real roots if a < – 4d)f(x) has three real roots if – 4 < a < 4Correct answer is option 'B,D'. Can you explain this answer?.
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