Question

Continuous differentiable?

The figure shows two parts of the graph of a continuous differentiable function `f` on [-10,4].
The derivative `f'` is also continuous.

(a) Explain why `f` must have at least one zero in [-10,4].

(b) Explain why `f'` must also have at least one zero in the interval [-10,4]. What are these zeros called?

(c) Make a possible sketch of the function with one zero of `f'` on the interval [-10,4]

Answer 1

If you need more details, let me know.

Explanation:

Part a

`0` is between `f(-10)` and `f(4)`, and `f` is continuous, so, by the intermediate Value Theorem, there is a `c` in `[-10,4]` at which `f(c) = 0`

Part b

`f` is continuos, so the Extreme Value Theorem guarantees an absolute maximum and an absolute minimum on the interval.

From the graph, we can see that the maximum is not at an endpoint, therefore, by Fermat's Theorem, the maximum occurs at a critical number for `f`. A critical number for `f` is a number in the domain of `f` at which the derivative either fails to exist or is equal to `0`.

We are told that `f` is differentiable on `[-10,4]`, so the derivative at the maximum must be `0`.

part c

Connect the endpoints with a smooth curve (no corner points or cusps) that has just one maximum. (One turning point)

Note

Given that the function is differentiable on `[-10,4]`, we can be certain that it is also continuous on `[-10,4]`.

Possibly the question=writer intended that we be given:

`f` is continuous on `[-10,4]` and differentiable on `(-10,4)`