Question

lf x, y are two real numbers such that $x^2+y^2=1$, then the maximum value of x+y is

• A. $\sqrt{2}$
• B. $\sqrt{5}$
• C. 2
• D. 6

Answer 1

The correct option is A $\sqrt{2}$

Let $x=cos\theta$ and $y=sin\theta$
Then, $f\left(\theta \right)=cos\theta +sin\theta$
$f^{\prime }\left(\theta \right)=-sin\theta +cos\theta$
For maxima or minima,
$f^{\prime }\left(\theta \right)=0$
$\Rightarrow \theta =\frac{\pi }{4}$
$f^{\prime \prime }\left(\theta \right)=-(cos\theta +sin\theta )$
$\Rightarrow f^{\prime \prime }\left(\frac{\pi }{4}\right)<0$
Hence, f has a maximum value at $\theta =\frac{\pi }{4}$
$f\left(\frac{\pi }{4}\right)=\sqrt{2}$