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By Lagrange’s mean value theorem which of the following statement is true:
(a) If a curve has a tangent at each of its points then there exists at least one-point C on this curve, the tangent at which is parallel to chord AB
(b) If f’(x) = 0 in the interval then f(x) has same value for every value of x in (a, b)
(a) If a curve has a tangent at each of its points then there exists at least one-point C on this curve, the tangent at which is parallel to chord AB
(b) If f’(x) = 0 in the interval then f(x) has same value for every value of x in (a, b)
- a)(a) alone is true
- b)(b) alone is true
- c)Both (a) and (b) are true
- d)Neither (a) nor (b) is true
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
By Lagrange’s mean value theorem which of the following statemen...
Concept:
Lagrange’s Mean Value Theorem:
If f(x) is real valued function such that –
Lagrange’s Mean Value Theorem:
If f(x) is real valued function such that –
- f(x) is continuous in the closed interval [a,b]
- (f(x) is differentiable in the open interval (a,b)
- f(a) ≠ f(b)
Then there exist at least one value x, c (a,b) such that –
Geometrical Interpretation:
Geometrical Interpretation:
- Between two points a and b, f(a) ≠ f(b) of the graph of f(x) then there exists one point where the tangent is parallel to the chord
Explanation:
(a) is true with reference to geometrical interpretations.
(b) is false, if f(x) has same value for every value of x, it will violate f(a) ≠ f(b).
(a) is true with reference to geometrical interpretations.
(b) is false, if f(x) has same value for every value of x, it will violate f(a) ≠ f(b).
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By Lagrange’s mean value theorem which of the following statemen...
Understanding Lagrange's Mean Value Theorem
Lagrange's Mean Value Theorem (MVT) states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point C in (a, b) such that the derivative at that point is equal to the slope of the secant line connecting points A and B.
Statement Analysis
- (a) If a curve has a tangent at each of its points, then there exists at least one point C on this curve, the tangent at which is parallel to chord AB:
- This statement accurately reflects the essence of the Mean Value Theorem. If the curve is differentiable (having a tangent at each point), then there is guaranteed to be a point C where the tangent line (slope) matches the slope of the secant line connecting points A and B. Therefore, this statement is true.
- (b) If f’(x) = 0 in the interval, then f(x) has the same value for every value of x in (a, b):
- This statement is misleading. If the derivative is zero everywhere in the interval, it implies that the function is constant; however, it does not necessarily mean that the function has the same value across the entire interval unless additional conditions are specified. Thus, this statement is false.
Conclusion
- Therefore, the correct answer is option 'A.' Only the first statement is true, as it aligns with the implications of Lagrange's Mean Value Theorem. The second statement does not hold without further context.
Lagrange's Mean Value Theorem (MVT) states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point C in (a, b) such that the derivative at that point is equal to the slope of the secant line connecting points A and B.
Statement Analysis
- (a) If a curve has a tangent at each of its points, then there exists at least one point C on this curve, the tangent at which is parallel to chord AB:
- This statement accurately reflects the essence of the Mean Value Theorem. If the curve is differentiable (having a tangent at each point), then there is guaranteed to be a point C where the tangent line (slope) matches the slope of the secant line connecting points A and B. Therefore, this statement is true.
- (b) If f’(x) = 0 in the interval, then f(x) has the same value for every value of x in (a, b):
- This statement is misleading. If the derivative is zero everywhere in the interval, it implies that the function is constant; however, it does not necessarily mean that the function has the same value across the entire interval unless additional conditions are specified. Thus, this statement is false.
Conclusion
- Therefore, the correct answer is option 'A.' Only the first statement is true, as it aligns with the implications of Lagrange's Mean Value Theorem. The second statement does not hold without further context.
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