II want to use the Intermediate Value Theorem and Rolle’s theorem to show that the graph of f

Mara Cook

Mara Cook

Answered question

2022-06-24

II want to use the Intermediate Value Theorem and Rolle’s theorem to show that the graph of f ( x ) = x 3 + 2 x + k crosses the x-axis exactly once, regardless of the value of the constant k.

I know I can use the intermediate value theorem, but I don't necessarily know how to show the change in a sign for two select inputs. any hero would be appreciated in that regard.

I also know the derivative of x 3 + 2 x + k is greater than zero, but what does that mean?

Answer & Explanation

benedictazk

benedictazk

Beginner2022-06-25Added 22 answers

Your function f ( x ) = x 3 + 2 x + k has derivative 3 x 2 + 2 which, as you say, is greater than 0. This means that f is strictly increasing. This results, for instance, from the mean value theorem. Now suppose f is 0 at two distinct points x and y, then f ( x ) = f ( y ). But we must have either x < y or y < x. We can assume x < y and then f ( x ) < f ( y ) since f is increasing.

As you suggest, we use the intermediate value theorem to demonstrate that there is at least one solution to the problem. For any fixed k we can choose x large enough such that x 3 + 2 x + k > 0. If we choose x large but negative we get x 3 + 2 x + k > 0. The intermediate value theorem now proves it.

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