Review question

# If $x^2+y^2\leq1$, when does $x+y$ have a maximum? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6645

## Question

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.

1. 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.

2. 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 $Q$?

3. 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 $Q$?