The correct choice is (iii), because at the inflection point the function changes its concavity, so the tangent line moves from being above the graph to be below the graph or vice versa, so it must cross the graph from one side to the other.
(i) is not correct. See the Questions B 01; Although the zero is closer than another zero to the initial point , but the newton method starting at finds the zero .
(ii) is not correct. The tangent line of at a point is . Assume that there is a tangent line which goes through . This means that there exists a point such that . However there's no such point because the maximum of is . This is a contradiction.
We find the maximum in the following way. Let . Then, its derivative is , which vanishes at . Indeed, these are critical points. so that the maximum in is achieved either the critical points or the end points . Since , the maximum is .
(iv) is not correct because we can take , , and ; Obviously, and are NOT differentiable at . but is a constant function and hence is differentiable at .
(Proof for (iii))
We need to show changes the sign around . Since , it is enough to show that is either increasing or decreasing around . (If it is increasing the sign is changing from negative to positive, otherwise it is from positive to negative.)
For this purpose, we calculate the derivative . Since changes the sign around , doesn't change its sign around . For example, in the case that change its sign from negative to positive around , then for sufficiently close with , and for sufficiently close with , .
Therefore, we prove that is either increasing or decreasing around .