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Q.
The minimum value of $f(x, y)=x^2-4 x+y^2+6 y$ when $x$ and $y$ are subjected to the restrictions $0 \leq x \leq 1$ and $0 \leq y \leq 1$, is
Application of Derivatives
Solution:
We have $ f ( x , y )= x ^2+ y ^2-4 x +6 y$
$\text { Let } ( x , y )=(\cos \theta, \sin \theta) \text {, then } \theta \in[0, \pi / 2] \text { and }$
$f ( x , y )= f (\theta)=\cos ^2 \theta+\sin ^2 \theta-4 \cos \theta+6 \sin \theta $
$ f ^{\prime}(\theta)=6 \cos \theta+4 \sin \theta>0 \forall \theta \in[0, \pi / 2] $
$\therefore f ^{\prime}(\theta) \text { is strictly increasing in }[0, \pi / 2] $
$\therefore f ^{\prime}(\theta)_{\min }= f (0)=1-4+0=-3 $