Question

Write sufficient conditions for a point x = c to be a point of local maximum.

Answer 1

We know that at the extreme points of a function f(x), the first order derivative of the function is equal to zero, i.e.
${\displaystyle f\prime \left(x\right)=0\text{at}x=c}$

${\displaystyle \Rightarrow f\prime \left(c\right)=0}$

Also, at the point of local maximum, the second order derivative of the function at the given point must be less than zero, i.e.
${\displaystyle f\prime \prime \left(c\right)<0}$