# What are the global and local extrema of #f(x) = 2x^3 - 5x +1# ?

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Problem:

What are the global and local extrema of

- The terms "local extrema" and "relative extrema" are often used interchangeably, however they can be used in specific and different ways, depending somewhat on the person using the terms.

**"local extrema" is often associated with finding extreme values over a specific, well defined open interval**, where as the term **"relative extrema" usually refers to extreme values found by using 1st and 2nd derivatives**, regardless of the actual domain of the function.

Assuming, the derivative method is intended for finding such values {and, I shall use the term "relative extrema"}:

**Setting # f'(x) = 0 # will yield critical points, to be explored further to determine their nature.**

**critical points**

**Evaluate # f''(x) # at x = 0 and for the above values to determine the nature of the cps.**

**so x = 0 is an inflection point.**

**Since this is positive,** the curve "holds water" here, so this location is a **relative minimum value.**

**Since this is negative,** the curve "spills water" here, so this location is a **relative maximum value.**

**Global max and min?**

Since,

**the global max and min are at positive and negative infinity, respectively.**

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