So my TA asked me a question the other day and it makes no sense to me. He says

Question

So my TA asked me a question the other day and it makes no sense to me. He says it is based off the Rolles Theroem, yet we never learned that yet (it isn't for another 2 or 3 chapters). Here was the question:

Let the f be a differentiable function with a max at x=a, show that f ' (a) = 0

We went about solving it like this:

f(a) = lim h-> 0  [ f(a+h) - f(a) ] / h

lim h->0-   [ f(a+h) - f (a) ] / h approaches infinity this it is greater than or equal to zero

lim h->0+ [ f(a+h) - f(a) ] / h approaches negative infinity thus it is less than or equal to zero

making the f'(x) = o

I don't understand how those two functions approach negative and positive infiinty. Please help me understand it, thanks.

Explanation / Answer

Maybe you copied this down wrong because neither of those would approach infinity.

If f has a max at a then f(a+h) < f(a) so f(a+h)-f(a) < 0 for small values of h (with h =/= 0).

For small values of h > 0, f(a+h)-f(a) < 0 and h>0 so [ f(a+h) - f(a) ] / h < 0. So the right-hand limit can only be negative or 0.

For small values of h < 0, f(a+h)-f(a) < 0 and h<0 so [ f(a+h) - f(a) ] / h > 0. So the left-hand limit can only be positive or 0.

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