How to solve an equation with a parameter using matan?

Question

Mathematics

Hello, I can't solve the following problem:

Find all values ​​of the parameter a for which the equation x^3-ax-1=0 will have a unique solution.

Because Since the task is in the topic of function extremes / increase and decrease, I suspect that it is solved through them. As far as I understand, you need to dig in the direction that if a given function has extrema, then they must be strictly less than zero, and if there are no extrema, then the function is strictly increasing => the only solution.

If you could give me some pointers on where to think to decide, I would be immensely grateful.

Answer 1

A cubic function with a first coefficient of 1, like yours, goes to minus infinity on the left and plus infinity on the right. It may not have extremums or have two - the maximum is to the left of the minimum.
If there are no extremums, there will be an increasing function, always with one real root. If there are extremums, then both of them must be less than 0, or both must be greater.
Here, take the derivative, find when it has roots and where they are. Substitute them into the original function and find the extrema - there will be expressions from a. Next, make up the conditions: either there are no roots, or they exist and the maximum is less than 0, or they exist and the minimum is greater than 0. Solve the inequalities and find the intervals for a.