A rectangle is to be inscribed under the arch of the curve $y = 4\cos(0.5x)$ from $x = \pi$ to $x = -\pi$. What are the dimensions of the rectangle with largest area, and what is the largest area?
Attempt: I have graph it and by looking at the graph, I can see the area of the inscribed rectangle is $A = (2x)(y)$. Where $2x$ is the total base, and y is the height.
Now I substitute $y = 4\cos(0.5x)$ on the $A = (2x)(y) = 2x[4\cos(0.5x)]$.
So $A = 8x\cos(0.5x)$. Then the derivative using the product rule is with respect to $x$ is
$$A' = 4\cos(x) - 4x\sin(0.5x)$$
However, when I tried to set the derivative equal to zero to find the critical numbers I don't know how to solve $\cos(x) = x\sin(0.5x)$. Can anyone please help me? I would really appreciate the help. Thank you.