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If a differentiable function f (x) has a relative minimum at x = 0, then the function y = f (x) + a x + b has a relative minimum at x = 0 for
- a)all b if a = 0
- b)all a > 0.
- c)all b > 0
- d)all a and all b
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If a differentiable function f (x) has a relative minimum at x = 0, th...
And y has a relative minimum at x = 0 if f ‘(0)+a = 0 . a =0.
Most Upvoted Answer
If a differentiable function f (x) has a relative minimum at x = 0, th...
Understanding Relative Minimums
When a differentiable function f(x) has a relative minimum at x = 0, it means that:
- The first derivative f'(0) = 0 (the slope is flat at this point).
- The second derivative f''(0) > 0 (the curve is concave up at this point).
This indicates that f(x) achieves a local minimum at x = 0.
Analyzing the Function y = f(x) + ax + b
We want to analyze the behavior of the function y = f(x) + ax + b at x = 0.
- First Derivative: The first derivative of y is:
- y' = f'(x) + a
At x = 0, since f'(0) = 0, we have:
- y'(0) = f'(0) + a = 0 + a = a
- Conclusion for Finding Minimum:
- For y to have a relative minimum at x = 0, we need y'(0) = 0.
- This implies that a must equal 0.
Effect of Adding b
The constant b does not affect the derivative of y. It merely shifts the graph vertically.
- Thus, regardless of the value of b, the condition for y to have a relative minimum at x = 0 solely depends on a being zero.
Conclusion
Therefore, the function y = f(x) + ax + b has a relative minimum at x = 0 for:
- All b if a = 0.
This confirms that option 'A' is correct.
When a differentiable function f(x) has a relative minimum at x = 0, it means that:
- The first derivative f'(0) = 0 (the slope is flat at this point).
- The second derivative f''(0) > 0 (the curve is concave up at this point).
This indicates that f(x) achieves a local minimum at x = 0.
Analyzing the Function y = f(x) + ax + b
We want to analyze the behavior of the function y = f(x) + ax + b at x = 0.
- First Derivative: The first derivative of y is:
- y' = f'(x) + a
At x = 0, since f'(0) = 0, we have:
- y'(0) = f'(0) + a = 0 + a = a
- Conclusion for Finding Minimum:
- For y to have a relative minimum at x = 0, we need y'(0) = 0.
- This implies that a must equal 0.
Effect of Adding b
The constant b does not affect the derivative of y. It merely shifts the graph vertically.
- Thus, regardless of the value of b, the condition for y to have a relative minimum at x = 0 solely depends on a being zero.
Conclusion
Therefore, the function y = f(x) + ax + b has a relative minimum at x = 0 for:
- All b if a = 0.
This confirms that option 'A' is correct.
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If a differentiable function f (x) has a relative minimum at x = 0, then the function y = f (x) + a x + b has a relative minimum at x = 0 fora)all b if a = 0b)all a > 0.c)all b > 0d)all a and all bCorrect answer is option 'A'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If a differentiable function f (x) has a relative minimum at x = 0, then the function y = f (x) + a x + b has a relative minimum at x = 0 fora)all b if a = 0b)all a > 0.c)all b > 0d)all a and all bCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a differentiable function f (x) has a relative minimum at x = 0, then the function y = f (x) + a x + b has a relative minimum at x = 0 fora)all b if a = 0b)all a > 0.c)all b > 0d)all a and all bCorrect answer is option 'A'. Can you explain this answer?.
If a differentiable function f (x) has a relative minimum at x = 0, then the function y = f (x) + a x + b has a relative minimum at x = 0 fora)all b if a = 0b)all a > 0.c)all b > 0d)all a and all bCorrect answer is option 'A'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If a differentiable function f (x) has a relative minimum at x = 0, then the function y = f (x) + a x + b has a relative minimum at x = 0 fora)all b if a = 0b)all a > 0.c)all b > 0d)all a and all bCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a differentiable function f (x) has a relative minimum at x = 0, then the function y = f (x) + a x + b has a relative minimum at x = 0 fora)all b if a = 0b)all a > 0.c)all b > 0d)all a and all bCorrect answer is option 'A'. Can you explain this answer?.
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