How to Prove: Fermat's Theorem

[@ln_cript]

beautifully and simply explained!

[@wishizuk]

Thanks for your kind words!

[@sciencedoneright]

Nice video! I always took the theorem for granted, but I guess the proof is indeed simple

[@wishizuk]

Thanks for your kind words! Yes, it is very simple. I hope that you will enjoy my other videos!

[@minamishi]

Thank you for the video but how do we know that the left hand limit is guaranteed to be equal to the right hand limit?

[@wishizuk]

If f'(c) exists, which means that the limit of the difference quotient exists, then the left-hand limit and the right-hand limit are equal; otherwise, the limit would not exist.

[@minamishi]

@@wishizuk I think what I was missing was. that a limit exists if and only if the one sided limits from the left and right are defined and equal

[@wishizuk]

@@minamishi Great! Glad to be able to help you find the missing piece!

[@kemalunal4776]

I don't understand why f(c+h)-f(c) can be zero. Can you explain? If h is zero it can be zero but then the denominator is zero.

[@wishizuk]

If f(x) = k, where is a constant, f(c+h) - f(c) = k - k = 0, so it can be zero, but that is not the point, though. The left-hand side is nonnegative, and the right-hand side in nonpositive, so the only possibility for f'(c) is 0.

[@danilojonic]

f(c) - f(c) = 0. h is not 0 but rather very small value approaching 0 therefore denominator is okay. Take a look back at f(c+h) - f(c) for the left example. Since h is 0 from the left (-0,000001 for example) then that means f(c + h) < f(c) and therefore top part is less than 0 and same analogy can be applied for the right part.