Maths A level question - help needed

Q wants point of inflection from the 1st equation. and exact co-ordinates.

I am confused on the differentiation part, can someone elaborate on this process please? on how the answer is formed - what it means.why is the change of root neccesary here?

Q wants point of inflection from the 1st equation. and exact co-ordinates.

I am confused on the differentiation part, can someone elaborate on this process please? on how the answer is formed - what it means.why is the change of root neccesary here?

A point of inflection occurs when a graph changes from convex to concave or vice versa, as you might know when the graph is convex the second derivative is above zero and when it is concave the second derivative is less then zero. Now since a point of inflection occurs when the graph changes between these two it means it occurs when the second derivative is equal to zero. So thats what theyve done in the first half, theyve set the second derivative equal to zero and found it. The next bit where they put x=0 and x=-5 in to the second derivative is just them checking if the graph has indeed changed from a convex to a concave or vice versa. as the second derivative can also be zero at a point that is not a point of inflection. So this step in the exams is something you should do because you can drop marks.

Yes thanks, i'll make sure more practice is needed this weekend/monday.
One part that is confusing, is differentiating the expression of xe and how they got dy/dx what is the rule if u know, and of course 2nd deritative just repeats that

no problem , they have used the product rule to differentiate it. You use the product role when there are two functions which are multiplying each other. such as this one, x times e^x/2 . x is function of x and e^x/2 is a function of x as well. You should look up the product rule there is also the quotient rule which is used when there are two functions being divided..