# Thread: Extrema on an Interval

1. ## Extrema on an Interval

The surface area of a cell in a honeycomb is

where h and s are positive constants and θ is the angle at which the upper faces meet the altitude of the cell. Find the angle θ (π/6 ≤ θ ≤ π/2) that minimizes the surface area S.

I started by taking the first derivative of S, but I was still left with multiple variables, even though h and s are constants. I'm not sure how to actually do this problem. Any help?

Thanks

2. Do you have to differentiate the equation implicitly?

3. I have now added a picture if anyone was confused.

4. Anybody know where I should start in solving this?

5. You should differentiate S with respect to theta, and then find the critical points. To discriminate between the different types of critical points you should look at the second derivative. Also don't forget to check the boundary points pi/6 and pi/2 even if the derivative might not vanish there.

6. The problem is that, for any $\theta$ or s, h can be anything. Making h shorter and shorter will make the total surface area less without changing $\theta$ or s. I presume, then, that you are asked to find the value of $\theta$ that will miminize surface area for any s and h. Treat s and h as constants.