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The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in the interval (1, 4) at a value
(rounded off to the second decimal place) of x equal to __________.
(rounded off to the second decimal place) of x equal to __________.
Imp : you should answer only the numeric value
Correct answer is '2.645'. Can you explain this answer?
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The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in t...
By Lagrange’s mean value theorem there exists C is (1, 4) such that
Lagrange’s means value theorem satisfied
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The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in t...
Given:
- Function f(x) = x^3 - 5
- Interval (1, 4)
To Find:
- The value of x at which the Lagrange mean-value theorem is satisfied
Solution:
The Lagrange mean-value theorem states that 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 value c in the interval (a, b) such that the derivative of f(x) at c is equal to the average rate of change of f(x) over the interval [a, b].
In other words, if f(x) is continuous and differentiable on the interval (a, b), then there exists at least one value c in the interval (a, b) such that:
f'(c) = (f(b) - f(a))/(b - a)
Applying the Lagrange mean-value theorem:
- The given function f(x) = x^3 - 5 is continuous and differentiable for all real numbers.
- The interval (1, 4) is an open interval.
- So, we can apply the Lagrange mean-value theorem to find the value of c in the interval (1, 4) at which f'(c) is equal to the average rate of change of f(x) over the interval (1, 4).
Calculating the average rate of change:
- f(1) = 1^3 - 5 = -4
- f(4) = 4^3 - 5 = 59
- The average rate of change of f(x) over the interval (1, 4) is (59 - (-4))/(4 - 1) = 21
Finding the value of c:
- Set f'(c) = 21
- Differentiating f(x) with respect to x, we get: f'(x) = 3x^2
- Set 3c^2 = 21 and solve for c:
3c^2 = 21
c^2 = 21/3
c^2 = 7
c = √7 or -√7
Choosing the appropriate value of c:
- Since the interval (1, 4) does not include negative values, we discard -√7 as a potential value of c.
- Therefore, the value of c in the interval (1, 4) at which the Lagrange mean-value theorem is satisfied is c = √7.
Rounded off value:
- Rounding off √7 to the second decimal place, we get 2.65.
- Therefore, the value of x at which the Lagrange mean-value theorem is satisfied is approximately 2.65.
- Function f(x) = x^3 - 5
- Interval (1, 4)
To Find:
- The value of x at which the Lagrange mean-value theorem is satisfied
Solution:
The Lagrange mean-value theorem states that 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 value c in the interval (a, b) such that the derivative of f(x) at c is equal to the average rate of change of f(x) over the interval [a, b].
In other words, if f(x) is continuous and differentiable on the interval (a, b), then there exists at least one value c in the interval (a, b) such that:
f'(c) = (f(b) - f(a))/(b - a)
Applying the Lagrange mean-value theorem:
- The given function f(x) = x^3 - 5 is continuous and differentiable for all real numbers.
- The interval (1, 4) is an open interval.
- So, we can apply the Lagrange mean-value theorem to find the value of c in the interval (1, 4) at which f'(c) is equal to the average rate of change of f(x) over the interval (1, 4).
Calculating the average rate of change:
- f(1) = 1^3 - 5 = -4
- f(4) = 4^3 - 5 = 59
- The average rate of change of f(x) over the interval (1, 4) is (59 - (-4))/(4 - 1) = 21
Finding the value of c:
- Set f'(c) = 21
- Differentiating f(x) with respect to x, we get: f'(x) = 3x^2
- Set 3c^2 = 21 and solve for c:
3c^2 = 21
c^2 = 21/3
c^2 = 7
c = √7 or -√7
Choosing the appropriate value of c:
- Since the interval (1, 4) does not include negative values, we discard -√7 as a potential value of c.
- Therefore, the value of c in the interval (1, 4) at which the Lagrange mean-value theorem is satisfied is c = √7.
Rounded off value:
- Rounding off √7 to the second decimal place, we get 2.65.
- Therefore, the value of x at which the Lagrange mean-value theorem is satisfied is approximately 2.65.
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The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in the interval (1, 4) at a value(rounded off to the second decimal place) of x equal to __________.Imp : you should answer only the numeric valueCorrect answer is '2.645'. Can you explain this answer?
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The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in the interval (1, 4) at a value(rounded off to the second decimal place) of x equal to __________.Imp : you should answer only the numeric valueCorrect answer is '2.645'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in the interval (1, 4) at a value(rounded off to the second decimal place) of x equal to __________.Imp : you should answer only the numeric valueCorrect answer is '2.645'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in the interval (1, 4) at a value(rounded off to the second decimal place) of x equal to __________.Imp : you should answer only the numeric valueCorrect answer is '2.645'. Can you explain this answer?.
The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in the interval (1, 4) at a value(rounded off to the second decimal place) of x equal to __________.Imp : you should answer only the numeric valueCorrect answer is '2.645'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in the interval (1, 4) at a value(rounded off to the second decimal place) of x equal to __________.Imp : you should answer only the numeric valueCorrect answer is '2.645'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The Lagrange mean-value theorem is satisfied for f (x) = x3 + 5, in the interval (1, 4) at a value(rounded off to the second decimal place) of x equal to __________.Imp : you should answer only the numeric valueCorrect answer is '2.645'. Can you explain this answer?.
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