NIMCET Previous Year Questions (PYQs) – Maxima And Minima – Page 1 of 2

$4{\sin }^{2}x+3{\cos }^{2}x=3({\sin }^{2}x+{\cos }^{2}x)+{\sin }^{2}x=3+{\sin }^{2}x\le 4$

$\sin \frac{x}{2}+\cos \frac{x}{2}\le \sqrt{2}$ with equality when ${\displaystyle \frac{x}{2}}={\displaystyle \frac{\pi }{4}}+2n\pi$

When ${\sin }^{2}x=1$ and $\sin \frac{x}{2}+\cos \frac{x}{2}=\sqrt{2}$, both maxima occur simultaneously.

Therefore, maximum value $=4+\sqrt{2}$

Answer: $\boxed{{\displaystyle 4+\sqrt{2}}}$ ✅

Compute partial derivatives: 

$f_x=2x-2,f_y=4y+4$.

Set $f_x=f_y=0\Rightarrow 2x-2=0,4y+4=0\Rightarrow (x,y)=(1,-1)$.

Hessian: $H=\left(\begin{array}{cc}2& 0\\ 0& 4\end{array}\right)$ (positive definite since $2>0$ and $det=8>0$).

Therefore, $(1,-1)$ is a strict (global) minimum. Also, by completing squares: $f(x,y)=(x-1{)}^2+2(y+1{)}^2-5\Rightarrow f_{min}=-5$ at $(1,-1)$.

Answer: Critical point $\boxed{{\displaystyle (1,-1)}}$; nature: $\boxed{{\displaystyle \text{minimum}}}$; minimum value $\boxed{{\displaystyle -5}}$.