h(x)= x + cos(ax), where a is a positive constant such that 0 < a < 4. For what values of a will h(x) have a relative maximum at x=1?

So I found the derivative and got h'(x) = 1- asin(ax). Then I plugged in 1 for x and set the derivative equal to zero, so I have 1 + asin(a)=0. I think this is right so far, I just don't know how to solve for a at this point. Thank you!!

You are on the right track! To solve the equation 1 + asin(a) = 0 for a, you have correctly set the derivative equal to zero.

Now, let's solve it step by step. Starting with the equation:

1 + asin(a) = 0

To isolate the term with a, subtract 1 from both sides:

asin(a) = -1

Next, divide both sides by a to get:

sin(a) = -1 / a

Now, you need to consider the range of possible values for a. It is given that 0 < a < 4, so the values of a lie between 0 and 4 exclusive.

The equation sin(a) = -1 / a only holds true for certain values of a. To find these values, you can draw the graph of y = sin(a) and y = -1 / a. The intersection points of these two graphs will give you the values of a that satisfy the equation.

Alternatively, you can use a calculator or graphing software to plot the graphs of sin(a) and -1 / a in the given range (0 < a < 4) and find the intersection points.

Keep in mind that the equation sin(a) = -1 / a might have multiple solutions within the given range. Once you have found those values of a, you can determine which ones satisfy the condition for h(x) to have a relative maximum at x = 1.