Question

Question #1214f

Answer 1

I. and III. must be true. II. can be false.

Explanation:

Statement I.

Note that `f(1) = f(3) = -2`.

Since `f(x)` is differentiable on `[1, 3]` it satisfies the conditions of the mean value theorem:

• It is continuous on `[1, 3]`

• It is differentiable on `(1, 3)`

The slope of a line through `(1, f(1))` and `(3, f(3))` is `0`, so by the mean value theorem there must be some `c in [1, 3]` such that `f'(c) = 0` as required.

Statement II.

Consider the function:

`f(x) = -6(x-2)^2+4`

This satisfies all of the conditions given by the question, but has zero derivative only at one point, the vertex of the parabola at `(2, 4)`

Statement III.

By the intermediate value theorem, or its simpler formulation, Bolzano's theorem, note that `f(x)` changes sign in `[1, 2]` and in `[2, 3]`. Since it is differentiable, it is continuous and so has a zero in both of those intervals.