## Function Extrema Lecture 7

### Topic: Sufficient condition for the existence of a function’s extremum (change in the sign of the derivative).

#### Summary

As we found out in the previous Lecture, just because the function’s derivative at a point is equal to 0 doesn’t necessarily mean that the function itself reaches an extremum at that point. So, here we’re going to talk about what conditions are **sufficient** for a function to reach an extremum at some point.

### Sufficient Conditions for the Existence of an Extremum

Assume that in some neighborhood of the point x_0, the function f \left(x \right) has a finite derivative f' \left( x \right):

- If in this neighborhood of x_0, to the left of x_0, the values of the function’s derivative are positive, and to the right of x_0 negative – then
**the function assumes a maximum at the point x_0** - If in this neighborhood of x_0, to the left of x_0, the values of the function’s derivative are negative, and to the right of x_0 positive – then
**the function assumes a minimum at the point x_0**

Indeed, according to the Lemma on the Monotonicity of Functions introduced in the previous Lecture, if the function’s derivative takes positive values, it means that the function

If, therefore, the derivative

The derivative to the left of

It’s clear that such a change always means the existence of a maximum at the point

THE END

In writing this post, I used…

1. “Differential and Integral Calculus. Volume I.” G.M. Fichtenholz. Published in 1966.

Click here to revisit the necessary condition for the existence of an extremum (previous Lecture) <–

Click to return to the lectures page on studying the variability of functions