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notice... the dog's pen perimeter, does not include the side that's bordering the garden's, since that side will use the heavy duty fence, instead of the light one
so, the sum of both of those costs, will be the C(x)
so, just take the derivative of it, and set it to 0 to find the extremas, and do a first-derivative test for any minimum
You have the following function:
Derivate implictly the previous expression, as follow:
Where you have used that:
Then, the implicit derivative of the given expression is:
Next, solve for y' as follow: