Question

While minimizing the function $f\left(x\right),$ necessary and sufficient condition for a point $x_0$ to be a minima are

• A. $f^{\prime }\left(x_0\right)>0$ and $f^{\prime \prime }\left(x_0\right)=0$
• B. $f^{\prime }\left(x_0\right)<0$ and $f^{\prime \prime }\left(x_0\right)=0$
• C. $f^{\prime }\left(x_0\right)=0$ and $f^{\prime \prime }\left(x_0\right)<0$
• D. $f^{\prime }\left(x_0\right)=0$ and $f^{\prime \prime }\left(x_0\right)>0$

Answer 1

The correct option is D $f^{\prime }\left(x_0\right)=0$ and $f^{\prime \prime }\left(x_0\right)>0$

$f\left(x\right)$ has a local minimum at $x=x_0$ if $f^{\prime }\left(x_0\right)=0$ and $f^{\prime \prime }\left(x_0\right)>0$