I am taking a calculus class and I've already asked for help but I didn't really get it 🙁
Can someone help me start this? Thanks!
Let $f(x)$ and $g(x)$ be functions with domain $(0,\infty).$ Suppose $f(x)=x^2$ and the tangent line to $f(x)$ at $x=a$ is perpendicular to the tangent line to $g(x)$ at $x=a$ for all positive real numbers $a$. Find all possible functions $g(x).$
Here's what I don't get/don't know what supposed to do: I tried graphing this on desmos and found it REALLY hard to find functions that would work and I'm guessing that the tangents would have to be close to (0,0). I'm assuming that g(x) is dependent on f(x). So, do I need to find functions for f(x) or is there something else?
Best Answer
First, let's find the slope of the tangent line to $f(x)$ at $x = a$. The derivative of $x^2$ is $2x$, so the slope of the tangent line is $2a$.
Because the two tangent lines are perpendicular, the slope of the tangent line to $g(x)$ is $-\frac1{2a}$. Basically, we want to find all the functions $g(x)$ can be for its derivative at $a$ to be $-\frac1{2a}$. So just take the antiderivative of $-\frac1{2a}$. We can replace $a$ with $x$ to get $-\frac12(x)^{-1}$, and the antiderivative is $$-\frac{\ln{x}}{2}$$ Adding $C$ as a constant (moving a function up and down will not change its derivative), we find that $$-\frac{\ln{x}}{2}+C$$ is the correct answer.