how many relative maxima and how many relative minima are there on the interval (1,10)? i don't see the way forward so i did some handwaving and said that the function in terms of maxima and minima is driven by cos(x). since cos(x) has 2 minima and one maximum on that interval, i claimed that the function also must have 2 minima and 1 max.

is there an analytical way to answer this?

Intermediate value theorem.

Extrema occur at points where , with maxima occurring at if the derivative is positive to the left of and negative to the right of , and minima in the opposite case.

So suppose you take two values . If it turns out that and , then the IVT guarantees the existence of some such that .

Choosing arbitrary values of won't guarantee that exactly one such exists, though. The function could easily oscillate several more times between and , intersecting the x-axis more than once, for example. This is where your suspicion can be applied. Knowing that for (approximately 1.57, 4.71, 7.85, respectively), you can use these values as reference points for computing the sign of the derivative.

When , you have . You know that for , and that as , . This means there must be some such that , and in particular, this value of is the site of a relative maximum.

You can use similar arguments to determine what happens at the other two suspected critical points.

What’s the question here? what have you been learning about?