Question

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.

Answer 1

A. Graph the curves y(x) = x + cos(¼x) and y(x) = x + cos(4x)

Answer 2

B. For what values of a will h(x) have a relative maximum at x = 1?

Answer 3

C. Which values of a make h(x) strictly decreasing? Justify your answer.

Answer 4

@perl

Answer 5

@phi

Answer 6

I need help with A and C. Already got B.

Answer 7

How far did you get ? Can you use Geogebra or other plotting tool?

Answer 8

or do you have to do it by hand ?

Answer 9

Didn't even think about using a graphing tool for part A. I'll do that now.

Answer 10

Alright I got part A now, can you help me with C?

Answer 11

I think we might need to use the work I did for part B so i'll post it.

Answer 12

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

Answer 13

strictly decreasing means as x increases f(x) gets smaller.

Answer 14

I don't see how to be decreasing when we are adding x

Answer 15

That's what I thought too, but how should I justify?

Answer 16

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.

Answer 17

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. )

Answer 18

So then it would only be 1.1 and 2.8?

Answer 19

and + not -

Answer 20

yes. But how did you find those values?

Answer 21

http://www.wolframalpha.com/input/?i=1%3D+a+sin%28a%29

Answer 22

oh, because it looks hard to solve (without the wolf)

Answer 23

exactly, that's why I used it

Answer 24

Alright, well thanks for your help!