Answer the following questions for the function f(x)= -6x - 9x + 540x - 1ő. a. Find formulas for r(x) and r"(x). r(x) = "(x) =O Enter f(x), f'(x), and f"(x) into your grapher to examine the table. b. The formula for the first derivative f (x) can be factored. Set f'(x) = 0 to find the two critical numbers. Hint: You can factor out - 18 from all terms in the formula for f (x). You can also s the table function on your calculator. The critical values are x= (Use a comma to separate answers as needed.) c. Use your table to complete the following. At the negative critical value listed in part b, what does your table tell you about the value of the second derivative? f' D =O(Type integers or simplified fractions.) %3D Fonsequently, what can be concluded about the graph of f? Select the correct choice below and, if necessary, fill in the answer boxes within your choice. OA. The graph of f is concave down and f has a relative minimum at ( I ). Uonh of f is concave up and f has a relative maximum at (.
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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