Question

What is the first and second derivative test?

Answer 1

The derivatives is used to find critical points of function, saddle point, maxima and minima of function.

First order test:

• To find the critical point of a function, we equate the first order derivative to zero that is $\frac{df}{dx}=0$.
• To find the maxima or minima point, we equate the first order derivative to zero that is $\frac{df}{dx}=0$.

Second order test:

• If the second order derivative at the critical point is less than zero, then the function has maxima at that critical point.
  $\frac{d^{2}f}{d{x}^2}<0$ at $x=a$, then the function $f$ has maxima at $x=a$.
• If the second order derivative at the critical point is greater than zero, then the function has minima at that critical point.
  $\frac{d^{2}f}{d{x}^2}>0$ at $x=a$, then the function $f$ has minima at $x=a$.