Verify Mean Value Theorem, if f (x) = x2 – 4x – 3 in the interval [a, b], where a = 1 and b =4.


Given: f(x) = x2 – 4x – 3 in the interval [1, 4]


Mean Value Theorem states that for a function f : [a, b] R, if


(a)f is continuous on [a, b]


(b)f is differentiable on (a, b)


Then there exists some c (a, b) such that


As f(x) is a polynomial function,


(a) f(x) is continuous in [1, 4]


(b) f'(x) = 2x – 4


So, f(x) is differentiable in (1, 4).



f(4) = 42 – 4(4) – 3 = 16 – 16 – 3 = -3


f(1) = 12 – 4(1) – 3 = 1 – 4 – 3 = -6



There is a point c (1, 4) such that f'(c) = 1


f'(c) = 1


2c 4 = 1


2c = 1+4 =5


c = 5/2 where c (1,4)


The Mean Value Theorem is verified for the given f(x).


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