In this problem, you will investigate the family of functions h(x) = x + cos(ax) , where a is a positive constant such that 0 < a < 4.

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In this problem, you will investigate the family of functions h(x) = x + cos(ax) , where a is a positive constant such that 0 < a < 4.

Mathematics
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A. Graph the curves y(x) = x + cos(¼x) and y(x) = x + cos(4x)
B. For what values of a will h(x) have a relative maximum at x = 1?
C. Which values of a make h(x) strictly decreasing? Justify your answer.

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I need help with A and C. Already got B.
  • phi
How far did you get ? Can you use Geogebra or other plotting tool?
  • phi
or do you have to do it by hand ?
Didn't even think about using a graphing tool for part A. I'll do that now.
Alright I got part A now, can you help me with C?
I think we might need to use the work I did for part B so i'll post it.
h'(x) = 1- a sin (ax); h'(1) = 0 h'(1)= 1 - a sin (a) = 0 1- a sin (a) = 0 1 = a sin a a is approx: +- 9.3 , +- 6.4, +- 2.8 , +- 1.1
  • phi
strictly decreasing means as x increases f(x) gets smaller.
  • phi
I don't see how to be decreasing when we are adding x
That's what I thought too, but how should I justify?
  • phi
strictly decreasing means that at every x the slope of the function is never 0 or positive. i.e. the slope is always negative. f'(x) = 1 - a sin(ax) and because the sin(ax) will be negative for 1/2 of its period, the -a sin(ax) adds to 1 and that is clearly not negative.
  • phi
btw, how did you solve a sin a = 1 for a ? (and you should restrict your a's to the interval: a is a positive constant such that 0 < a < 4. )
So then it would only be 1.1 and 2.8?
and + not -
  • phi
yes. But how did you find those values?
http://www.wolframalpha.com/input/?i=1%3D+a+sin%28a%29
  • phi
oh, because it looks hard to solve (without the wolf)
exactly, that's why I used it
Alright, well thanks for your help!

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