Question

What is the second derivative test used for?

Answer 1

Sometimes first derivatives fail to determine the maxima and minima of the function.

So, the second derivative test is done to determine the maxima and minima of the function.

• The function $\mathrm{f}\left(\mathrm{x}\right)$ is maximum at the point where the first derivative of the function is equal to zero and the second derivative is negative.
  That is $f\text{'}\text{'}\left(x\right)<0$ at $x=a$, then $f\left(x\right)$ has local maxima at $x=a$.
• The function $\mathrm{f}\left(\mathrm{x}\right)$ is minimum at the point where the first derivative of the function is equal to zero and the second derivative is positive.
  That is $f\text{'}\text{'}\left(x\right)>0$ at $x=a$, then $f\left(x\right)$ has local minima at $x=a$.
• To find the saddle point, we use partial second-order derivatives.