# Algebraic derivative solution

I have a function that I need to find $$x$$ where the maximum of $$y$$ occurs.

$$f(x)= 0.00394\log(1+75x)+0.02335\log(1+25x)+0.97271\log(1-x).$$

Using this graph, I find $$x$$ where maximum of $$y = -0.002179$$.

How to go about finding solving this algebraically? I am not a math heavy guy and never studied calculus but maybe I can follow along with the right direction.

This MSE question is similar to the one brought up here. I have modified that example for my use case.

If you could help cheers! If not have a great day :)

Algebraically it might be somewhat annoying because of the logs, but in general, if your function is continuously differentiable, you should try to solve the equation $$f'(x)=0$$ to find critical points. The intuition behind this is that when a function reaches either a maximum or a minimum, it will change direction, so its slope will change sign and therefore go through 0. Luckily, your function contains only three logs, so you can compute its derivative as $$\frac {a_1}{1+b_1x}+\frac {a_2}{1+b_2x}+\frac {a_3}{1+b_3x}=0$$ Multiplying both sides by $$(1+b_1x)(1+b_2x)(1+b_3x)$$ will give you a sum of quadratics which you can then solve with the quadratic formula. After that just plug the solutions into your original function and compare which one is largest. Alternatively you could try computing the second derivative and using the second derivative test, but that seems like it is more trouble than it is worth.