Saturday, 24 January 2015

calculus - Show that the Mean Value Theorem does not apply to f(x)=x2 on (1,1)



Show that the Mean Value Theorem does not apply to f(x)=x2 on (1,1). Is this a contradiction to the Mean Value Theorem?



My solution so far:



f(1)f(1)=f(c)(1(1))11=2f(c)f(c)=0




Since f(x)=2x3f(c)=2c3



f(c)=2c3=0.



This is not satisfied by any value c so I think I've now got the first part of the task. However, I'm not quite sure how do I figure if this contradicts with the Mean Value Theorem. I know that the MVT tells us that if f is continuous on [a,b] and differentiable on (a,b), there is a point c(a,b) such that f(b)f(a)=f(c)(ba) but I don't know how to apply it here.


Answer



The mean-value theorem only applies for continous functions. But x2=1x2 is not defined at x=0 , the singularity is not even removeable.


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