Struggling with finding the maximum for this function:
$$f(x)=\Bigl(\theta-\Bigl(\frac{\mu}{p-x}\Bigr)^a\Bigr)·x$$
where $\theta>0, \mu>0, p>0, \alpha>1$.
Wolfram Alpha gives me the differential but then for some strange reason it can't solve it. I tried a few other free resources, all of them either refuse to solve it or e.g. stop at a certain step.
Here is the shape of the function just for illustration:
Edit: just noticed the question was closed because of lack of context. This comes from economics.
The function is supposed to be used for revenue maximization. is a price variation for a product and is a number of buyers willing to pay a given price. It's based on the hypothesis that price tolerance is a function of wealth distribution. You can see Pareto distribution in the formula, flipped and displaced; then it is multiplied by the price () to give the value of revenue. The idea is that you can field-test presumably 3 price points, get the curve and then find your maximum for revenue.
If I did not miscalculate ... Edit: of course I did :) thanks @mr_e_man
$$f'(x) = \Bigl(\theta-\Bigl(\frac{\mu}{p-x}\Bigr)^a\Bigr) + xa\Bigl(\frac{\mu}{p-x}\Bigr)^{a+1} (-1/\mu) = \theta - \Bigl(\frac{\mu}{p-x}\Bigr)^a \Bigl(1+\frac{xa}{p-x}\Bigr)$$