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Problems for "The Mean Value Theorem"

Problems for "The Mean Value Theorem"

Problems for "The Mean Value Theorem"

Problems for "The Mean Value Theorem"

Problems for "The Mean Value Theorem"

Problems for "The Mean Value Theorem"


In problems 1-3, for each of the following functions f defined on [a, b] find the c on [a, b] such that

f'(c) =    

Problem : 1) f (x) = x 2 - 4x on [2, 4]


f'(c) = = 2  
2c - 4 = 2  
c = 3  

Problem : 2) f (x) = sin(x) + cos(x) on [0, 4Π]


    f'(c) = = 0  
    cos(x) - sin(x) = 0  
    x = ,,, or  

Problem : 3) f (x) = on [1, 2]


f'(c)   =  
    = -  
-   = -  
c   = ±  

Problem : 4) On the interval [-5,5], there is no point at which the derivative of f (x) =|x| is equal to zero, even though f (- 5) = f (5) . Is this a contradiction of Rolle's theorem?

No, it isn't a contradiction, since this function is not differentiable on the entire interval (- 5, 5) .

Problem : Find the number c that satisfies Rolle's theorem for f (x) = sin(x) on the interval [0, Π] .

(sin(x))' = cos(x)
cos(x) = 0 at x =