Find max or min:
y = x * cosx
dy/dx = x(-sinx) + cosx
= 0 for a max/min
xsinx = cosx
sinx/cosx = 1/x
tanx = 1/x
No direct algebraic method to solve this, so we have to resort to other means
let's look at the graph by Wolfram
A preliminary look at y = xcosx shows that there would be an infinite number of max/mins and there is symmetry in the curve.
Looking at the tanx = 1/x shows this:
http://www.wolframalpha.com/input/?i=plot+tanx++%3D+1%2Fx
I see two intersection points between -1.5 and 1.5
and hovering my cursor over the intersection point I get x = appr .860333 and -.860333
So we have a local max at x = .86033 of .5611
and a local min at x = -.86033 of -.5611
there is another max/min for x between 3 and 4, and between -3 and -4
http://www.wolframalpha.com/input/?i=+tanx++%3D+1%2Fx+for+x+%3D+.001+to+5
and if x = 3.4256, then y = -3.2884
these x's produce max/mins of y = xcosx
http://www.wolframalpha.com/input/?i=+tanx++%3D+1%2Fx