Maths question ln

Hi, I was trying to simplify this but I’m getting the wrong answer. I’m not sure where I have gone wrong though, please could someone check it? I have attached my working below.
https://ibb.co/JsGRSzF
Thanks!

It would help to see the original question and your writing isnt the clearest, but near the bottom sqrt(k) seems to become k (both times) on the penultimate line?

The first part of the question is a specific value of k (-3) and is meant to get you thinking about the derivative etc. For that value, the gradient is always > 0, so increasing, which it obviously isnt in (ii) so they want you to think about what values of k might correspond to a function which has a minimum (zero gradient) and has "u" shape and whats the range of g()
So if you differentiate the g(x), can you find the values of x which correspond to a minimum. Assuming you have the minimum when e^(4x) = k or e^(2x) = sqrt(k), you can sub that into g() and pretty much write down gmin, rather than going through your working and its 2sqrt(k) as mentioned in the previous post.
But its less about an algebra slog and more about thinking what you need to do and as its only 5 marks, there shouldnt really be an algebra slog in the first place.

Thank you, I tried your method and it works! But, I'm still unsure where my algebra has gone wrong in the previous post, it may be useful for future questions.

You have
g(x) = sqrt(k) + k / sqrt(k)
Really the left hand side isnt a function of x any more (it kind of shows youre not sure what youre calculating, so gmin(k)) but you seem to multiply the right hand side only by sqrt(k) to get 2k. However its an equation and you have to maintain balance so if you multiplied the right by sqrt(k) youd have to do the same to the left to get
g(x)*sqrt(k) = 2k
which is the same as g(x) = 2sqrt(k).
I guess youre not treating it as an equation and the "= " literally means that both lines are the same length or both sides balance / same value. You cant make one line longer arbitrarily.

oh, i see where i went wrong, thank you so much 🙂