How can i find where two functions meet to have the same gradient?

Question

I have two functions, ax^2 and clog(x)-2. How can I find the point where they can meet at a gradient that is the same.

Answer 1

Assuming you're asking when the two functions $ax^2$ and $c \text{log}(x) - 2 $ have the same gradient for any given real number $a$ and $c$, you can start of by finding the derivative of both functions, i.e.

$$ \frac{d}{dx}\left[ax^2\right] = 2ax $$

$$ \frac{d}{dx}\left[c \ \text{log}(x) - 2 \right] = \frac{c}{x} $$

Then to find the point where they are equal, you can equate them:

$$ 2ax = \frac{c}{x} $$

And now you can solve for $x$ (Note: you may need to assert conditions on $a$ and $c$ if you'll be taking square roots!).