This Question: 1 pt 23 of 30 For the function below, find a) the critical numbers; b) the open intervals where the function is increasing; and c) the open intervals where it is decreasing. f(x) = 12x° - 45x² - 216x + 5 a) Find the critical number(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The critical number(s) is/are (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no critical numbers. b) List any interval(s) on which the function is increasing. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The function is increasing on the interval(s) (Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) O B. The function is never increasing. c) List any interval(s) on which the function is decreasing. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function is decreasing on the interval(s) (Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) O B. The function is never decreasing. Click to select and enter your answer(s). Save for Later
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.
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