if f'(x)=sin(lnx), how many relative extrema does f have in the internal xe(0,2]? Examine this carefully?

Question

This is for AP Calculus AB. Any answers/work is greatly appreciated ASAP

Answer 1

Are you sure that you've expressed this problem correctly?

Oh, wait, is there a problem with the bold font making the start of the question hard to read? I initially thought that the FUNCTION f(x) was sin (ln x), but since there's an EXTREMELY MINUTE BLIP on the apparent "f," perhaps it's f'(x)?!

I see, the problem is even there with that latter f primed, in regular, plain text. So I take it we have:

f primed (x) = sin (ln x). (This has more promising possibilities.)

Not quite sure what is meant by "relative extrema"; are they any different to (plain) extrema? Assuming not, we need to find ALL the places in the range (0, 2] where sin (ln x) is giving us zero.

[Afterthought: I guess that "relative extrema" gives a nod to the fact that they all crowd together as x ---> 0.]

An extremum in f (x) occurs when f (primed) (x) = 0. Now the first non-zero argument "nzarg1" for which sin (nzarg1) itself = 0 is nzarg1 = pi = 3.14159265358... . But if ln x = pi, x = e^pi, and THAT x-value is WAY BEYOND the upper limit 2 of the range given. So THAT'S not relevant.

Turning our attention to smaller values of x where sin (ln x) = 0, the next one corresponds to having ln x = 0, or x = 1. That IS in the range (0. 2] The next value of x for which we'd find sin (ln x) = 0 would be when ln x = - pi. (Then sin (ln x) = sin (- pi) = 0.) This is the point x = e^(- pi).

Now, I get it; we're on a roll. f (primed) (x) = sin (ln x) will be zero for the set of x such that ln x = 0, - pi, - 2 pi, -3 pi, ... etc., without limit, which gives the following set of x's:

x = 1, e^(- pi), e^(-2 pi), e^(- 3 pi), ... with general term e^(- n pi) for n = 0, 1, 2, 3, ... (n ---> infinity).

So it isn't possible to express the number of extrema (relative extrema?) in (0, 2] as a finite number; it's an infinite number. (There must be some mellifluous phrase like "... an infinite number with the same cardinality as the positive integers." But reading about things with such fancy phrases lies almost 50 years in my past, back in Starfleet Academy days, so you're really on your own there.)

I hope that this helps.

Live long and prosper.

POSTSCRIPT: You might want to note this extra little point of detail:

Whether the extrema will be MINIMA or MAXIMA of the original function f(x) depends upon whether f (primed) (x) changes from -ve to +ve, or from +ve to -ve, respectively, on passing through the relevant value of x in the +ve direction. For x = 1, ln (x) goes from -ve to +ve, so does sin (ln x), so that's a MINIMUM for f(x). That situation will recur for x = e^(- 2 pi): ln x passes through -2 pi , and the behaviour of sin (ln x) there is the same as at x = 1. This holds for all even multiples of (- pi).

Conversely, the opposite behaviour happens for the odd multiples of (- pi): for example, as x goes through e^(- pi), ln x goes from less than (- pi) to more than (- pi), and sin (ln x) there changes from +ve to -ve; so that value gives a MAXIMUM.

So: the even multiples of (- pi) in the exponent of 'e' give you MAXIMA, the odd ones give you MINIMA.