# Which function, yellow, blue, red shows a minimum and which shows a maximum ? Describe, in your own words, what a maximum or minimum of a function is.

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Maximum and minimum are known as the **extrema of a function**. They can be called as **local or relative extrema** that exist as highest or lowest function value within a part of domain of the function. If it is the highest or smallest function value (y-value) for “all x” in the domain of *f*, then it is known as **absolute extrema.** The **maximum **represents the largest value of the function or highest y-value at the peak of a curve (looks like a *hill top*). While the **minimum** represent the smallest value of the function at the bottom of the curve (looks like a *valley*). By visual inspection, the **blue and yellow curves shows minimum **and the** red curve shows a maximum.**

Aside from visual inspection, we can evaluate the maximum and minimum based on the following conditions.

**Definition of Maximum and Minimum:**

The function of *f* has a maximum at “c” that exist within the interval of (x1,x2) implies f(c) ≥ f(x).

The function of *f* has a minimum at “c” that exist within the interval of (x1,x2) implies f(c) ≤ f(x)

Note “c” represent a critical x-value in the function of f.

**First derivative test:**

For function of *f* that has a critical number “c” within a continuous and open interval, the f(c) can be classified as:

If f’(x) changes from **negative to positive** at “c”, then f(c) is a relative minimum.

If f’(x) changes from **positive to negative** at “c”, then f(c) is a relative minimum.

If f’(x) does not change sign at c, then f(c) is not a relative extrema.

Relative minimum: **f’(x) <0** at the left side of c and **f’(x)>0 **at the right side of c.

Relative maximum: **f’(x) >0** at the left side of c and **f’(x)<0 **at the right side of c.

**Second Derivative test:**

For a function that has f’(c) =0, the second derivative can be used to test if c is at the relative maximum or relative minimum.

If f”(c) >0, then f(c) is a relative minimum.

If f”(c)<0, the f(c) is a relative maximum.