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While minimizing the function $f(x)$, necessary and sufficient conditions for a point, $x_0$ to be a minima are:

  1. $f’(x_0) > 0$ and $f’’(x_0)=0$
  2. $f’(x_0) < 0$ and $f’’(x_0)=0$
  3. $f’(x_0) = 0$ and $f’’(x_0)<0$
  4. $f’(x_0) = 0$ and $f’’(x_0)>0$
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