Question

A) find the intervals on which f is increasing or decreasing.
B) find the local maximum and minimum values of f
C) Find the intervals of concavity and the inflection points.

Where....

Answer 1

\[f(x)= \sin (x)+\cos (x), 0\le x \le2\pi\]

Answer 2

i guess what i really need help with is the critical points in the first derivative text and in the second derivative test.

Answer 3

\[f'(x) = \cos(x) - \sin(x) = Rcos(x + \alpha)\] for some alpha.

Expanding:

cos(x) - sin(x) = Rcos(a)cos(x) - Rsin(a)(sin(x)

=> R cos(a) = R sin (a) = 1 => tan a = 1 => a = pi/4.

R^2(sin^2(a) + cos^2(a)) = 2 => R = sqrt(2)

Gogogogogog

Answer 4

hmm thanks i'm pretty sure i need to approach that differently

Answer 5

You think I'm wrong?

Answer 6

f'(x) = 0
iff cos(x-pi/4) = 0
iff x-pi/4 = pi/2 , 3pi/2
iff x = 3pi/4, 7pi/4
You have your critical values

Answer 7

the alpha and r is was confused me. i've never had to use that in calculus
sorry. didnt mean to like offend or anything

Answer 8

Well, my method gets you the critical values of d/dx f(x).

And your method doesn't exist.

Answer 9

Maybe you should ask Leibniz instead ;-)

Answer 10

Just realised I should take off pi/4 - sorry, I'm getting old. It was a copying error, rather than a mathematical one (because I don't make them).

Critical values are pi/4, 5pi/4

Oh, and on the subject of your precious leibniz :
http://imgs.xkcd.com/comics/newton_and_leibniz.png

Answer 11

ha that was funny.

i still dont understand where the R or alpha come from. i'm sorry i havent had trig in a really long time so these trig equations always throw me off

Answer 12

In general, you can rewrite

m * cos(x) + n sin(x) as R cos (x ± a) (or R sin (x ± a) because of how they expand. if you work out R and, more importantly for this problem, a, you can use it to work out the minimum/maximum/critical etc values. It's done because it's easier to work with an equation with once trig function than 2:

Answer 13

There is probably some other way to do this, but I can't think of it right now, so it's probably not too good.

Answer 14

You could use the same method on the original equation (not the derivative) - if you are good at sketching, it's quicker.