Find any relative extrema of the function. List each extremum along with the x-value at which it occurs. Identify intervals over which the function is increasing and over which it is decreasing. Then sketch a graph of the function. f(x) = - x - 9x2 Describe any relative extrema. Select the correct choice below and, it necessary, fill in the answer box(es) within your choice. O A. The relative maximum point(s) is/are and there are no relative minimum points. (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) O B. The relative minimum point(s) is/are and the relative maximum point(s) is/are. (Simplify your answers. Type ordered pairs, using integers fractions. Use a comma to separate answers as needed.) OC. The relative minimum point(s) is/are and there are no relative maximum points. (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) O D. There are no relative minimum points and there are no relative maximum points. Identify any intervals over which the function increasing or decreasing. Select the correct choice below and fill in the answer box(es) within your choice. O A. The function f(x) is increasing over the interval(s) and decreasing over the interval(s). (Type your answers in interval notation. Use integers or fractions for any numbers in the expressions. Use a comma to separate answers as needed.) O B. The function f(x) is increasing over the interval(s) and is not decreasing anywhere. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) OC. The function f(x) is decreasing over the interval(s) interval notation. Use integers and is not increasing anywhere. fractions for any numbers in the expression. Use a comma to separate answers as needed.) (Type your answer Choose the correct graph below.

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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.