Answer :
By the Mean Value Theorem, there exists a value c in the interval [1,4] such that f'(c) is equal to the average rate of change of f on the interval [1,4], which is (f(4) - f(1))/(4-1).
We can start by computing f(4) and f(1):
f(4) = 5cos(2(4^2)) + ln(4+1) - 3 = -0.841 + 1.609 - 3 = -1.232
f(1) = 5cos(2(1^2)) + ln(1+1) - 3 = 2.531 - 0.693 - 3 = -1.162
Then, we can compute the average rate of change:
(f(4) - f(1))/(4-1) = (-1.232 - (-1.162))/3 = -0.023
To satisfy the conclusion of the Mean Value Theorem, we need to find a value c in the interval [1,4] such that f'(c) = -0.023. From the given expression for f'(x), we can see that there is no value of c that satisfies this equation, since f'(x) can never be negative. Therefore, there is no value of c that satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,4].
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