Question

What is true for the differentiable function?

For the differentiable function f(x) on the interval [1, 3]
f(1) = -2, f(2) = 4, f(3) = -2
Which of the following must be true?
I) There must be at least one point on the interval where f'(x) is zero
II) There must be at least two points on the interval where f'(x) is zero
III) There must be at least two points on the interval where f(x) is zero

Answer 1

Statements `I` and `III` are true. See explanation.

Explanation:

Proof of `I`

`f(2)>f(1)`, so `f(x)` is increasing in `(1;2)`,

`f(3) < f(2)` , so `f(x)` is decreasing in `(2;3)`

This means that `f(x)` has a maximum at `x=2`, so `f'(2)=0`

Proof of `III`

The function is differentiable in `[1;3]`, so it is also continuous on this interval.

`f(1)=-1` and `f(2)=4`, so there is at least one point where every value from `-1` to `4` is taken, so there is at least one zero in this interval.

The same reasoning can be repeated for interval `[2;3]`, so there is also at least one zero in `[2;3]`.

This concludes that there are at least 2 zeros in the interval `1;3]`.