Can somebody help me with this question from the 2015 Independent Trial Paper?

[joey9911]

Worth 3 marks:

Find the set of values of x for which the function f(x) = xe^(-x) is both decreasing and concave down.

Thank you!

[RuiAce]

[joey9911]

Thank you so much RuiAce! Perfect step-by-step explanation!

I have just one more question (from the same paper), since this one involves a graph, I'm going to screenshot and attach it to this post. I did (i), the answer I got was A(-1,3). I just need help with (ii)!

[RuiAce]

Thank you so much RuiAce! Perfect step-by-step explanation!

I have just one more question (from the same paper), since this one involves a graph, I'm going to screenshot and attach it to this post. I did (i), the answer I got was A(-1,3). I just need help with (ii)!

Yep. It's hard to explain over the internet (it's analysed through the definition of the integral as an area, and subtraction of areas) but the overall effect is that it will always give the area between the curves.

[joey9911]

Thank you so much again!

That's exactly what I thought of doing, although I thought that was wrong because I somehow believed that the area under the points between A and O would have been included in my final answer. Does subtracting the curve from the line when integrating only give me the area of the shaded region?

[joey9911]

Yep. It's hard to explain over the internet (it's analysed through the definition of the integral as an area, and subtraction of areas) but the overall effect is that it will always give the area between the curves.

Ahh! Understood Thank you so much!!! (Btw, what happened to the working out? xD - If it's deleted, it's fine!)

[RuiAce]

Oh my goodness. So sorry haha somehow I clicked edit post when I didn't intend to!

I'll just leave this here for reference

[jazza47]

Just in case you prefer my working out:)