# What is the first derivative test for critical points?

Apr 22, 2018

If the first derivative of the equation is positive at that point, then the function is increasing. If it is negative, the function is decreasing.

#### Explanation:

If the first derivative of the equation is positive at that point, then the function is increasing. If it is negative, the function is decreasing.
http://mathworld.wolfram.com/FirstDerivativeTest.html

Suppose $f \left(x\right)$ is continuous at a stationary point ${x}_{0}$.

1. If ${f}^{'} \left(x\right) >$0 on an open interval extending left from ${x}_{0} \mathmr{and} {f}^{'} \left(x\right) < 0$ on an open interval extending right from ${x}_{0}$, then $f \left(x\right)$ has a local maximum (possibly a global maximum) at ${x}_{0}$.

2. If ${f}^{'} \left(x\right) < 0$ on an open interval extending left from ${x}_{0} \mathmr{and} {f}^{'} \left(x\right) > 0$ on an open interval extending right from ${x}_{0} , t h e n f \left(x\right)$ has a local minimum (possibly a global minimum) at ${x}_{0}$.

3. If ${f}^{'} \left(x\right)$ has the same sign on an open interval extending left from ${x}_{0}$ and on an open interval extending right from ${x}_{0} , t h e n f \left(x\right)$ has an inflection point at ${x}_{0}$.

Weisstein, Eric W. "First Derivative Test." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FirstDerivativeTest.html