Question

2. 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.



B. For what values of a will h(x) have a relative maximum at x=1
C. For what value(s) of a will h(x) have an inflection point at x = 1.
D. Which values of a make h(x) strictly decreasing? Justify your answer.

Answer 1

@peachpi

Answer 2

B. Take the derivative, set it equal to 0, plug in 1 for x, and solve for a.
h'(x) = 1 - a sin (ax)
h'(1) = 1 - a sin (a) = 0
Solve for a

C. Take the 2nd derivative, set it equal to 0, plug in 1 for x, solve for a.
h"(x) = -a² cos (ax)
h"(1) = -a² cos (a) = 0
Solve for a

D. I don't think there are any. The cos (ax) part oscillates so it's neither strictly increasing nor strictly decreasing. If you eliminate the oscillating part by making a = 0, the function becomes h(x) = x + 1, which is strictly increasing.

Answer 3

I'm having trouble solving for a @peachpi

Answer 4

B.0.84?
C. 1?