Submitted by alexisA on Fri, 06/08/2018 - 00:18
Forums:
In the Hints for the question, the method for finding the upper bound on z^3 in part (ii) is given as using the fact that 1/3(4kz-z^4) is a square number. In the solutions it says that this is found by comparison with part (i), where the lower bound is found by 1/3(4z^3 -k^3).
I don't really understand how we go from the statement in part (i) to the statement in part (ii), it seems like they are, if equivalent in form, just arbitrarily chosen and I don't really get the methodology for deriving it as needed for the question.
Could someone help explain?
Thanks
Comparisons
When the solutions say "Comparison with part (i)" it really means try and deduce that something is a perfect square.
The difference between parts (i) and (ii) is that in the first one $x+y=k$ and in the second $x+k=z^2$ so in a sense $k$ is being replaced by $z^2$. In the first part you considered $\frac {4z^3-k^2}3$, and if we use $k=z^2$ in this we get $\frac {4 \times z^2 \times z - k^2}3=\frac {4kz-z^4}3$.
Does this help?