Question

Check my answers?

Answer 1

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.

My answers:
B. 0.84
C.1

Answer 2

Work for B.
h(x)=x+cos(ax)
h′(x)=1−asin(ax)
h′(1)=1−asin(a)=0
1−asin(a)=0
1=asin(a)

Answer 3

Work for C.
h"(x) = -a² cos (ax)
h"(1) = -a² cos (a) = 0
acos(a)=0

Answer 4

For Part B,
we can divide through by a,\[\large\rm \frac{1}{a}=\sin(a)\]I think you need a calculator to proceed further...
Anyway, I graphed 1/x and sinx separately,
https://www.desmos.com/calculator/k50bkjc4a9
and you can see that they intersect at x=1.114 and x=2.773.

But again, I'm not sure how to come up with these values without a calculator.

Answer 5

For part C, your work looks good so far.
\(\large\rm -a^2cos(a)=0\)
Divide through by -a^2,
\(\large\rm cos(a)=0\)

This corresponds to an angle value of pi/2, ya?
Not 1.

Answer 6

Oh oops thanks Zep!