Let \(Q\) denote the quarter-disc of points \((x,y)\) such that \(x\geq 0\), \(y\geq 0\) and \(x^2+y^2\leq1\) as drawn in Figures A and B below.

- On the axes in Figure A, sketch the graphs of \[x+y=\frac{1}{2}, \qquad x+y=1, \qquad x+y=\frac{3}{2}.\]

What is the largest value of \(x+y\) achieved at points \((x,y)\) in \(Q\)? Justify your answer.

What do we observe in Figure A after drawing the three graphs? How are they related?

- On the axes in Figure B, sketch the graphs of \[xy=\frac{1}{4}, \qquad xy=1, \qquad xy=2.\] What is the largest value of \(x^2+y^2+4xy\) achieved at points \((x,y)\) in \(Q\)? What is the largest value of \(x^2+y^2-6xy\) achieved at points \((x,y)\) in \(Q\)?

Is this part of the question similar to the previous part in any way?

How could we use the diagram to solve the problem?

*‘If the largest values of two functions, say \(f\) and \(g\), are \(\alpha\) and \(\beta\), and \(f+g\) can take the value \(\alpha+\beta\), then the largest value of \(f+g\) is \(\alpha+\beta\).’*

Is this logic okay?

What similar statement can we make about the largest value of \(f-g\)?

- Describe the curve \[x^2+y^2-4x-2y=k\] where \(k>-5\). What is the smallest value of \(x^2+y^2-4x-2y\) achieved at points \((x,y)\) in \(Q\)?

How else can we write \(x^2+y^2-4x-2y=k\)? What curve does this look like?