Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 8 / b / Solution

The sine addition formula gives $(\sin x+\cos x)/\sqrt{2}=\sin (x+\pi /4)$. On the requested interval the shifted angle runs from $0$ to $\pi /2$, so the graph rises from $(-\pi /4,0)$ through $(0,1/\sqrt{2})$ to $(\pi /4,1)$.

Its derivative $\cos (x+\pi /4)$ is nonnegative and reaches zero only at the right endpoint. Its second derivative is $-\sin (x+\pi /4)\le 0$, so the rising arc is concave downward and ends with a horizontal tangent.