Originally Posted by

**ThomasNewton12** Some of the posters on this forum have been thoroughly impressive, so I'm hoping someone can help me out. I apologize for the presentation not being more professional, was getting "[LaTeX ERROR: Unknown error]" when trying to use Latex. Here's the problem as it was presented:

Since if f(x) = x^{2} then f'(x) = 2x and f''(x) = 2 it seems clear that if f''(x) > 2, then f(x) > x^{2} . For the purposes of this problem we are to prove this using the mean value theorem. This was my approach:

Consider the interval [0,x] for some x > 0. Using the MVT there exists some c such that f''(c) = f'(x) - f'(0) / x -0 = f'(x) / x. This is > 2 by assumption. Multiplying by x yields f'(x) > 2x.

Next, again by MVT there exists some d such that f'(d) = f(x) - f(0) / x - 0 = f(x) / x. This is > 2x by the previous statement. Multiplying by x yields f(x) > 2x^{2} .

However, I've been instructed this is incorrect. Apparently I'm misunderstanding the statements I'm making and that last line should be f(x) > 2cx but I'm not entirely sure why. Any assistance in explaining where I went wrong and/or how to approach this proof correctly would be greatly appreciated. Thanks in advance!