If $x+y=k, x0, y0$, then $x^2+y^2$ is minimum, if

Question

If $x+y=k, x>0, y>0$, then $x^2+y^2$ is minimum, if

Aicharya · Accepted Answer

$ \begin{gathered} x+y=k \ \Rightarrow y=k-x, x>0, y>0 \end{gathered} $ Consider $ \begin{aligned} & A=x^2+y^2=x^2+(k-x)^2=2 x^2-2 k x+k^2 \ & \frac{d A}{d x}=4 x-2 k=0 \Rightarrow x=\frac{k}{2} \ & \frac{d^2 A}{d x^2}=4>0 \end{aligned} $ $	herefore$ At $x=y=\frac{k}{2}$, we have Minimum vale of $x^2+y^2$

If $x+y=k, x>0, y>0$, then $x^2+y^2$ is minimum, if

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