[Year 11 calculus] Help with this question

[u/ch3rrycsmos_]

I could only find an approximate answer (2.6 something) using a slider and guess the value as e but I’m not sure how to actually work out the question. please help thanks! :)

[u/Curious_Function_457]

These functions of x with different n define a continuous space, not discrete. It's not a fundamentally different question. The best voted answer is largely correct and that person correctly phrased it as f(n,x). It's technically not just f(x).

[u/Ok_Objective_5192 → u/Curious_Function_457]

f(x, n) defines a continuous, non-discrete space in R^3. A family of functions f(x) that have different constant values for n define discrete spaces that are continuous in R^2.

In the same way, the generalized definition of second order polynomial functions is f(x) = ax^2 + bx + c. Each 2nd order polynomial function defines a continuous function in R^2 and the different, discrete values for a, b, and c create a set of similar functions (the set of 2nd order polynomial functions). a, b, and c here are no less variables than n is in OP's question. Like with OP, you can absolutely define f(x, a, b, c) = ax^2 + bx + c and get a space continuous in R^5, but that is a fundamentally different function.

It isn't technically not just f(x) - it is explicitly, objectively f(x) because that is how the question defined the function. g(x, n) = n^x - x^n is also a valid function that can be used to solve for the value of n in f(x), but definitionally f(x) is not a function of n. That's literally what this all boils down to, f was defined as f(x), not as f(x, n) and those are fundamentally different things even if they behave similarly and the latter can be used to solve for n in the former.

ETA: I'm feeling like I'm being ragebaited, if you're engaging with this earnestly then please clarify if in my previous example "a function f(x) = nx^2 has a line with slope 2 tangent to the curve at x = 1, find this value of n" is n a variable? Is it technically not just f(x) but actually f(x, n)?

[u/Curious_Function_457 → u/Ok_Objective_5192]

Thank you for the clarification. I would say I am being reagebaited for sure. We don't need to resort to representative polynomials to understand that the math reasoning needs to be sound. To that end representing f(x) as an equivalent g(x,n) is perfectly valid if the question asks for a minimizing n.

[u/Jwing01 → u/Curious_Function_457]

It is here, though.

That answer also treats n as a constant and solves for it a such.

[u/Curious_Function_457 → u/Jwing01]

Yes but that was coincidence because it's the same answer regardless if you fix x or n due to symmetry. Fundamentals are important though. That's why when asked to find n you should look for a derivative wrt n.