1A graphical method can be much simpler.
take P and Q as the co-ordinate axis, then the constraint imlies that P and Q must lie on the circumference of a circle of radius 1.
Now, let P+Q=c
which in fact is a straight line having a P-intercept of c units.
It can be seen that c is maximum, with P and Q lying on the circle, happens only when the line P+Q=c touches the circle.
where P=Q (by symmetry)
or P=1/√2
and c=√2
