a. Find the open interval(s) on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. a. On what open interval(s), if any, is the function increasing? Select the correct choice below and, if necessary, fil in the answer box(es) to complete your choice. O A. The function is increasing on the open interval(s) (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed) O B. The function is never increasing. On what open interval(s), i any. is the function decreasing? Select the correct choice below and, if necessary. fil ithe answer box(es) to compleote your choice. O A. The function is decreasing on the open interval(s) (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) OB. The function is never decreasing b. Find each local maximum, if there are any. Select the correct choice below and, if necessary. fil inthe answer box(es) to complete your choice. (Simplify your answers. Type exact answers, using radicals as needed.) O A. The function has a local maximum value at three values of x. In increasing order of x-value, the maximum values are f(O-L0=| O B. The function has a local maximum value at one value of x. The maximum value is f( ) = OC. The function has a local maximum value at two values of x. In increasing order of x-value, the maximum values are f OD. There are no local maxima and and Find each local minimum, if there are any. Select the correct choice below and, if necessary. fi in the answer box(es) to complete your choice. (Simplity your answers. Type exact answers, using radicals as needed.) DA. The function has a local minimum value at three values of x. In increasing order of x-value, the minimum values are f() = O B. The function has a local minimum value at two values of x. In increasing order of x-value, the minimum values are f( - and f( O. The function has a local minimum value at one value of x. The minimum value is 1()- and OD. There are no local minima. If the function has extreme values, which of the extreme values, i any, are absolute? Select the correct choice below and fill in any answer boxes within your choice. (Simplify your answers. Type exact answers, using radicals as needed. Use a comma to separate answers as needed.)

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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.