Examine whether The Mean Value Theorem(M.V.T) is applicable to the following functions for the interval in which they are defined.   







SOLUTION: the function f(x)=x³-10x²-8x+1for 0≤x≤2 has interval [0,2] to verify the Mean Value Theorem we use the test of Mean Value Theorem which is 







Where a=0 and b=2 and the derivative of f(x) is
f'(x)=3x²-20x-8  and f'(x)=f'(c) 
f(0)=(0)³-10(0)²-8(0)+1=1
f(2)=(2)³-10(2)²-8(2)+1=-47











But f'(x)=f'(c) this implies that  f'(c)=3c²-20c-8
And since f'(c)=24 then 3c²-20c-8=-24
3c²-20c-8+24=0 hence c=5.74 and c=0.93

(ii)  the functions 







Does not satisfy the MVT as f(0) in the both functions does not exists and hence the both functions does not satisfy the  MVT.