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Geometrically the Mean Value theorem ensures that there is at least one point on the curve f(x) , whose abscissa lies in (a, b) at which the tangent is
- a)Parallel to the x axis
- b)Parallel to the y axis
- c)Parallel to the line joining the end points of the curve
- d)Parallel to the line y = x
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Geometrically the Mean Value theorem ensures that there is at least on...
GMVT states that for a given arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.
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Geometrically the Mean Value theorem ensures that there is at least on...
Understanding the Mean Value Theorem
The Mean Value Theorem (MVT) is a fundamental result in calculus that relates the behavior of a function on an interval to its derivative.
Statement of the Mean Value Theorem
- If a function f(x) is continuous on the 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:
f'(c) = (f(b) - f(a)) / (b - a)
Geometric Interpretation
- The expression f'(c) represents the slope of the tangent line to the curve at the point (c, f(c)).
- The fraction (f(b) - f(a)) / (b - a) represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).
Why is the Correct Answer 'C'?
- Parallel to the Line Joining the End Points: The slope of the secant line is the average rate of change of the function over the interval [a, b].
- The Mean Value Theorem guarantees that at least one point c exists where the instantaneous rate of change (the derivative) is equal to this average rate of change.
- Thus, the tangent line at point c is parallel to the secant line joining the end points (a, f(a)) and (b, f(b)), confirming option 'C' is correct.
Why Other Options are Incorrect
- Option A: A tangent parallel to the x-axis implies a zero slope, which is not guaranteed.
- Option B: A tangent parallel to the y-axis is not possible in the context of real-valued functions.
- Option D: A tangent parallel to y = x implies a slope of 1, which is also not guaranteed by the MVT.
In summary, the Mean Value Theorem ensures that there is a point where the tangent is parallel to the line that joins the endpoints of the curve, validating option 'C'.
The Mean Value Theorem (MVT) is a fundamental result in calculus that relates the behavior of a function on an interval to its derivative.
Statement of the Mean Value Theorem
- If a function f(x) is continuous on the 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:
f'(c) = (f(b) - f(a)) / (b - a)
Geometric Interpretation
- The expression f'(c) represents the slope of the tangent line to the curve at the point (c, f(c)).
- The fraction (f(b) - f(a)) / (b - a) represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).
Why is the Correct Answer 'C'?
- Parallel to the Line Joining the End Points: The slope of the secant line is the average rate of change of the function over the interval [a, b].
- The Mean Value Theorem guarantees that at least one point c exists where the instantaneous rate of change (the derivative) is equal to this average rate of change.
- Thus, the tangent line at point c is parallel to the secant line joining the end points (a, f(a)) and (b, f(b)), confirming option 'C' is correct.
Why Other Options are Incorrect
- Option A: A tangent parallel to the x-axis implies a zero slope, which is not guaranteed.
- Option B: A tangent parallel to the y-axis is not possible in the context of real-valued functions.
- Option D: A tangent parallel to y = x implies a slope of 1, which is also not guaranteed by the MVT.
In summary, the Mean Value Theorem ensures that there is a point where the tangent is parallel to the line that joins the endpoints of the curve, validating option 'C'.
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Geometrically the Mean Value theorem ensures that there is at least one point on the curve f(x) , whose abscissa lies in (a, b) at which the tangent isa)Parallel to the x axisb)Parallel to the y axisc)Parallel to the line joining the end points of the curved)Parallel to the line y = xCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Geometrically the Mean Value theorem ensures that there is at least one point on the curve f(x) , whose abscissa lies in (a, b) at which the tangent isa)Parallel to the x axisb)Parallel to the y axisc)Parallel to the line joining the end points of the curved)Parallel to the line y = xCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Geometrically the Mean Value theorem ensures that there is at least one point on the curve f(x) , whose abscissa lies in (a, b) at which the tangent isa)Parallel to the x axisb)Parallel to the y axisc)Parallel to the line joining the end points of the curved)Parallel to the line y = xCorrect answer is option 'C'. Can you explain this answer?.
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